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# BASICAPPLIEDSTATISTICS STAT0200

Pitt

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This 152 page Class Notes was uploaded by Josefa Cartwright Jr. on Monday October 26, 2015. The Class Notes belongs to STAT0200 at University of Pittsburgh taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/229432/stat0200-university-of-pittsburgh in Statistics at University of Pittsburgh.

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Lecture 3 Designing Studies Focus on Experiments uDefinitions uRandomization uControl uBIind Experiment nPitfalls uSoecific 39 Desiqns 2 mm mm mm swim shims mm We aw 7mm C 2007 Nancy Pfenning Looking Back Review El 4 Stages of Statistics I Data Production I Displaying and Summarizing I Probability I Statistical Inference mmmmnw mm Eiementaivs tstics mmmmw mm m2 Looking Back Review El 2 Types of Study Design I Observational study record variables values as they naturally occur El Drawback confounding variables due to self assignment to explanatory V ues El Example Men who drink beer are more prone to lung cancer than those who drink red wine what is the confounding variable here I Experiment researchers control values of varia El If well designed provides more convincing evidence of causation 2mm mnwmm amnuwsmsm mm We gimme m De nitions Elementary Statistics Looking at the Big Picture El Factor an explanatory variable in an experiment El Treatment value of explanatory variable imposed by researchers in an experiment A control group individuals receiving no treatment or baseline treatment may be included for comparison If individuals are human we call them subjects 2mm mm mm ammw Statstics mm mm aw mm m C 2007 Nancy Pfenning Example Randomized controlled experiment and ot ers to 1 Population assignments 2 mm mm mm Eiemenhw shims mm tithe aw 7mm El Background To test if sugar causes hyperactivity researchers assign some children to low h39h evels of sugar consumption Sample El Question What is the advantage of random Example Randomized controlled experiment El Background To test if sugar causes hyperactivity researchers randomly assign some children to low and others to high levels of sugar consumption El Response mumm mm Eiementaivstatstics makinva heaiv mm m Experiment vs Observational Study Populatio n a 2mm mm Eiemenhw shims mm tithe aw 7mm In an experiment researchers decide who has low sugar intake L and who has high H Sample Sugar intake as not yet been determined Researchers assign sugar intake L or H Experiment vs Observational Study In observational study individuals have already chosen low L or high H sugar intake Population Sample Researchers make no changes to sugar intake mumm mm Eiementaivstatstics makinva heaiv We L12 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Randomization at rst or second stage Consider two selection issues in our sugar hyperactivity experiment El What individuals are included in the study El Who consumes low and high amounts of sugar At which stage is randomization important 2mm mnwmm amnuwsmsm mm tithe swim a u Randomization at First or Second Stage Individuals studied may be volunteers Otherwise noncompliance may be an issue Population Sample Assignment to sugar L or H must be random Volunteering which treatment to get is not OK mmmmnm mm ammmysmm makmva heaiv mm L111 Must an experiment have a control group Recall our de nition El Experiment researchers manipulate explanatory variable observe response Thus experiment may have no control group El if all subjects must be treated El if simulated treatment is risky El if the experiment is poorly designed As long as researchers have taken control of the explanatory variable it is an experiment 2mm mnwmm amnuwsmsm mm tithe swim L112 Elementary Statistics Looking at the Big Picture De nitions Three meanings of control El We control for a confounding variable in an observational study by separating it out El Researchers control who gets what treatment in an experiment by making the assignment themselves ideally at random El The control group in an experiment consists of individuals who do not receive a treatment per se or who are assigned a baseline value of the explanatory variable mmmmnm mm ammmysmm inakmvanheaiv mm mm C 2007 Nancy Pfenning Doubleblind experiments Two pitfalls may prevent us from drawing a conclusion of causation when results of an experiment show a relationship between the socalled explanatory and response variables El If subjects are aware of treatment assignment El If researchers are aware of treatment assignment 2 mm mm mm swim shims mm tithe aw 7mm De nitions El The placebo effect is when subjects respond to the idea of treatment not the treatment itself El A placebo is a dummy treatment El A blind subject is unaware of which treatment heshe is receiving El The experimenter effect is biased assessment of or attempt to in uence response due to knowledge of treatment assignment El A blind experimenter is unaware of which treatment a subject has received mmmmnm mm armaments makmva heaiv mm Example Subjects not blind El Background Suppose after children are randomly assigned to consume either low or high amounts of sugar researchers nd proportion hyperactive is greater for those who consumed higher amounts El Question Can we conclude sugar causes hyperactivity swim shims mm tithe aw 7mm Elementary Statistics Looking at the Big Picture Example Subjects not blind El Background Suppose after children are randomly assigned to consume either low or high amounts of sugar researchers nd proportion hyperactive is greater for those who consumed higher amounts El Response El Improvement mmmmnm mm armaments makmva heaiv mm C 2007 Nancy Pfenning 3 1 Example Experimenters not blind Example Experimenters not blind El Background Suppose after children are randomly El Background Suppose after children arerandomly assigned to diets sweetened either arti cially or with ass1g ed t0 dletS sweetened miller a l CIauY 0 W1th sugar researchers nd proportion hyperactive rs sugar researchers find proportion hyperactive is greater for those who consumed sugar El Response greater for those who consumed sugar I Question Can we conclude sugar causes hyperactivity More problematic if responses are assessed Improvement c 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 i9 c 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 2i 2 2 Best evidence of causation Other pitfalls in experimentation In general conclusions of causation are most I Lack of realism lack of ecological validity COHVlnClng 1f a relatlonShlp has been I Hawthorne effect subjects behave differently established in a randomized controlled because of awareness of participation in doubleblind experiment experiment A Closer Look In the original studies reporting a relationship D Noncomphance between sugar and Imperaetivity Conducted in the 1970 I Treatments unethical experimenters may have been airvare thlie children 39s diet when they assessed bEhGVlOI randomized controlledsingleblineb U Treatments Impracticalimposs1ble t0 Impose Lilam studies since then havefailed to establish a relationship c 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 22 C 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 23 Elementary Statistics Looking at the Big Picture 5 C 2007 Nancy Pfenning Example Hawthorne e eet lack of realism Example Hawthorne e eet lack of realism El Background Suppose researchers want to determine if TV makes people snack more I While study participants are presumably wailing to be interviewed half are assigned to a room with a TV on and snacks the other half to a room with no TV and snacks See if those in the room with TV consume more snacks El Question If participants in the room with TV snack more can we conclude that in general people snack more when they watch TV 2mm mnwmm amnuwsmsm mm tithe swim ma El Background Suppose researchers want to determine if TV makes people snack more I While study participants are presumably waiting to be interviewed half are assigned to a room with a T Von and snacks the other half to a room with no TV and snacks See if those in the room with T Vconsume more snacks El Response if people suspect they re observed TV and snacking habits different in contrived setting 2mm mm mm ammw Statstics mm um aw We ma Example Non compliance in experiment Example Non compliance in experiment El Background To test if sugar causes hyperactivity researchers randomly assign 50 children to low and 50 to high levels of sugar consumption 20 drop out of each group For remaining children 30 in each group suppose proportion hyperactive is substantially greater in the highsugar group El Question Can we conclude sugar causes hyperactivity 2mm mnwmm amnuwsmsm mm tithe swim L127 El Background To test if sugar causes hyperactivity researchers randomly assign 50 children to low and 50 to high levels of sugar consumption 20 drop out of each group For remaining children 30 in each group suppose proportion hyperactive is substantially greater in the highsugar group El Res onse 7 makes treatment and control groups different in ways that may affect response 2mm mm mm ammw Statstics mm um aw mm mm Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Another awed experiment El Background To test if stuttering is a learned rather than inborn trait a researcher in Iowa in 1939 randomly assigned subjects to I Control 11 orphans in ordinary speech therapy I Treatment 11 orphans badgered and interrupted in sessions with speech therapist Of the 11 in treatment group 8 became stutterers Question What s wrong with this experiment El 2mm mnwmm amnuwsmgm mm mm swim mu Example Another awed experiment El Background To test if stuttering is a learned rather than inborn trait a researcher in Iowa in 1939 randomly assigned subjects to I Control 11 orphans in ordinary speech therapy I Treatment 11 orphans badgered and interrupted in sessions with speech therapist Of the 11 in treatment group 8 became stutterers El Response 2mm mm mm ammw Statstics mm um aw mm L132 Examples Treatments impossibleimpractical El men get married sooner promoted quicker and earn higher wages El There is a link betweenand low SOC 00110111 C status 111 women Height is impossible to control Weight is dif cult to control Socioeconomic status is too costly to control 2mm mnwmm amnuwsmgm mm mm swim in Elementary Statistics Looking at the Big Picture Modi cations to randomized experiment El Blocking Divide rst into groups of individuals who are similar with respect to an outside variable that may be important in relationship studied El Paired design Randomly assign one of each pair to receive treatment the other control Before and a er is a common paired design Looking Back block71g M to experimentation as Strati cation is to sampling 2mm mm mm ammw Statstics mm um aw mm L134 C 2007 Nancy Pfenning Example Blocked experiment Example Blocked experiment El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Question How to incorporate blocking if researcheis suspect location plays a role in relationship between type of sunscreen and amount of time spent in sun 2mm mnwmm amnuwsmgm mm tithe swim ms El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Response 2mm mm mm ammw Statstics mm um aw mm L137 Example Paired experiment Example Paired experiment El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Question How to incorporate paired design if researchers suspect location plays a role in relationship between type of sunscreen and amount of time spent in sun 2mm mnwmm amnuwsmgm mm tithe swim ma El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Response 2mm mm mm ammw Statstics mm um aw mm mu Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Paired experiment The paired design helps to ensure that treatment and control groups are as similar as possible in all other respects so that if their responses dijfkr we have evidence that the treatment is responsible 2mm mnwmm amnuwsmms mm tithe swim L141 Example Combining Paired and Two Sample Designs El Background Studies often randomly assign one group to a placebo and the other to a drug Responses to the variable of interest are assessed before and after a period of time then compared to see benefits or side effects El Question What aspect of the design is twosample and What aspect is paired 2mm mm mm ammw Stalstics mm um aw mm m 44 Example Combining Paired and Two Sample Designs El Background Studies often randomly assign one group to a placebo and the other to a drug Responses to the variable of interest are assessed before and after a period of time then compared to see benefits or side effects El Response twosample paired 2mm mnwmm amnuwsmms mm tithe swim m as Elementary Statistics Looking at the Big Picture Lecture Summary Experiments De nitions Randomization 2 stages ofselection Control group Blind study design I Subjects blind to avoid placebo elfect DUDE I Researchers blind to avoid experimenter elfect Other pitfalls of experiments lack of realism Hawthorne effect noncompliance unethical or impractical treatment B U Speci c experimental designs I Blocked Paired ortwosample 2mm mm mm ammw Stalstics mm um aw We L147 Lecture 13 Finding Probabilities Basic Rules uProbability Definition and Notation Basic Rules ulndependence Sampling V th Replacement 2 mm mm mm Eiemenhw mm mm We aw 7mm C 2007 Nancy Pfenning Looking Back Review El 4 Stages of Statistics I Data Production discussed in Lectures 14 I Displaying and Summarizing Lectures 512 I Probability El Random Variables El Sampling Distributions I Statistical Inference mmmmnm mm amnmysmms wwwmaiv mm m 2 Four Processes of Statistics Population 1 PRODUCE DATA Sample Amime we Assume we only know what39s true for the sample what can we in fer about the larger population Eiemenhwstahshcs mm We swim L113 De nitions Elementary Statistics Looking at the Big Picture El Statistics I Science of producing summarizing drawing conclusions from data I Summaries about sample data El Probability I Science dealing with random behavior Chance of happening mmmmnm mm amnmysmms wwwmaiv mm m 5 C 2007 Nancy Pfenning Example Ways to Determine a Probability El Background Some probability statements Probability of randomly chosen card a heart is 025 Probability of randomly chosen student in a class getting A is 025 according to the professor Probability of candidate being elected according to an editorial is 025 El Question Are these probabilities all determined in the same way 2mm mnwmm amnuwsmgm mm mm swim m a Example Ways to Determine a Probability El Background Some probability statements Probability of randomly chosen card a heart is 025 Probability of randomly chosen student in a class getting A is 025 according to the professor Probability of candidate being elected according to an editorial is 025 El Response These three probabilities are determined in different ways mumm mm ammwsmm makmvanhea v mm L117 De nition El Probability chance of an event occurring determined as the I Proportion of equally likely outcomes comprising the event39 or I Proportion of outcomes observed in the long run that comprised the event39 or I Likelihood of occurring assessed subjectively 2mm mnwmm amnuwsmgm mm mm swim m a Example Three Ways to Determine aProbability Elementary Statistics Looking at the Big Picture El Background Some probability statements Probability of randomly chosen card a heart is 025 5 Probability of randomly chosen student in a class getting A is 025 according to the professor Probability of candidate being elected according to an editorial is 025 El Question Is each determined as a I Proportion of equally likely outcomes Or I Proportion of longrun outcomes observed Or I Subjective likelihood of occurring mumm mm ammwsmm tmkmvameaiv mm L119 C 2007 Nancy Pfenning Example Three Ways to Determine a Probability El Background Some probability statements 1 Probability of randomly chosen card a heart is 025 El Response I Proportion of equally likely outcomes I Proportion of longrun outcomes observed I Subjective likelihood of occurring 2mm mnwmm amnuwsmsm mm tithe swim in 11 Example Three Ways to Determine aProbability El Background Some probability statements 1 Probability of randomly chosen card a heart is 025 2 Probability of randomly chosen student in a class getting A is 025 according to the professor El Response I Proportion of equally likely outcomes I Proportion oflongrun outcomes observed I Subjective likelihood of occurring mmmmnm mm Eiementaivstatstics makmvanheaiv mm mm Example Three Ways to Determine aProbability El Background Some probability statements 1 Probability of randomly chosen card a heart is 025 2 Probability of randomly chosen student in a class getting A is 025 according to the professor 5 Probability of candidate reelected according to an editorial is 025 Notation El Response I Proportion of equally likely outcomes g x um I Proportion oflongrun outcomes observed npi u39anx mg I Subjective likelihood of occurring 1quot We adhere W or amnuwsmsm mm tithe swim in 15 Use capital letters to denote events in probability Hevent of getting a heart Use P to denote probability of event PH probability of getting a heart Use not A to denote the event that an event A does not occur mmmmnm mm Eiementaivstatstics damning mm m a Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Basic Probability Rules To stress how intuitive the basic rules are we I Begin with example for which we can intuit the solution I State general rule based on solution I Apply rule to solve a second example Example I rituitirig Permissible Probabilities Rule Looking Ahead This process will be used to establish all the rules needed to understand behavior of random variables in general sampling distributions in particular so we have theory needed to perform inference 2mm mnwmm amnuwsmsm mm We swim m 7 El Background A sixsided die is rolled once El Questions What is the probability of getting a nine What is the probability of getting a number less than nine mmmmnm mm ammwsmm makmva heaiv mm ma Example I rituiting Permissible Probabilities Rule Permissible Probabilities Rule El Background A sixsided die is rolled once El Responses PN PL 2mm mnwmm amnuwsmsm mm We swim Lam Elementary Statistics Looking at the Big Picture The probability of an impossible event is 0 the probability of a certain event is l and all probabilities must be between 0 and l mmmmnm mm ammwsmm lnakmva heaiv mm m 21 Example Applying Permissible Probabilities ule C 2007 Nancy Pfenning El Background Consider the values l 01 01 10 El Question Which of these are legitimate probabilities 2mm mnwmm amnuwsmm mm tithe swim L522 Example Applying Permissible Probabilities ule El Background Consider the values l 01 01 10 El Response 2mm mm mm ammw Stalstics mm um aw mm L524 Example Intuiting SumtoOne Rule El Background Consider the roll of a sixsided die El Question What do we get if we sum the probabilities ofrolling a l a 2 a 3 a 4 a 5 and a 6 7 2mm mnwmm amnuwsmm mm tithe swim L1125 Example Intuiting SumtoOne Rule Elementary Statistics Looking at the Big Picture El Background Consider the roll of a sixsided die El Response 2mm mm mm ammw Stalstics mm um aw mm L1127 C 2007 Nancy Pfenning SumtoOne Rule The sum of probabilities of all possible outcomes in a random process must be 1 2mm mnwmm amnuwsmsm mm mm swim m 2s Example Applying SumtoOne Rule El Background A survey allows for three possible responses yes no or unsure We let PY PN and PU denote the probabilities of a randomly chosen respondent answering yes no and unsure respectively El Question What must be true about the probabilities PY PN and PU 2mm mm mm ammw Statstics mm um aw mm m 29 Example Applying SumtoOne Rule El Background A survey allows for three possible responses yes no or unsure We let PY PN and PU denote the probabilities of a randomly chosen respondent answering yes no and unsure respectively El Response Looking Back This rule con rms that sum of areas of pie charl s slices must be I or 100 2mm mnwmm amnuwsmsm mm mm swim m 1 Example Intuiting Not Rule Elementary Statistics Looking at the Big Picture El Background A statistics professor reports that the probability of a randomly chosen student getting an A is 025 El Question What is the probability of not getting an A 2mm mm mm ammw Statstics mm um aw mm m 32 C 2007 Nancy Pfenning Example Intuiting Not Rule Not Rule El Background A statistics professor reports that the probability of a randomly chosen student getting an A is 025 El Response Looking Back Alternatively since A and notA are the only possibilities according to the Sum to One Rule we must have 025Pnot AI so Pnot A 1 0250 75 2mm mnwmm amnuwsmgm mm tithe swim m 34 For any event A Pnot AlPA Or we can write PAlPnot A 2mm mm mm ammw Statstics mm um aw mm m 35 Example Applying Not Rule Example Applying Not Rule El Background The probability of a randomly chosen American owning at least one TV set is 098 El Question What is the probability of not owning any TV set 2mm mnwmm amnuwsmgm mm tithe swim m as El Background The probability of arandomly chosen American owning at least one TV set is 098 El Response Pnot TV 2mm mm mm ammw Statstics mm um aw mm m as Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example I rituitirig Non Overlapping Or Rule El Background A statistics professor reports that the probability of a randomly chosen student in her class getting an A is 025 and the probability of getting a B is 030 El Question What is the probability of getting an A or a B 2mm mnwmm amnuwsmsm mm mm swim m 39 Example I rituiting Non Overlapping Or Rule El Background A statistics professor reports that the probability of a randomly chosen student in her class getting an A is 025 and the probability of getting a B is 030 El Response 2mm mm mm ammw Statstics mm um aw mm m 41 Example When Probabilities Can t Simply be aided El Background A statistics professor reports that the probability of a randomly chosen student in her class getting an A is PA025 and the probability of being a female is PFO60 El Question What is the probability of getting an A or being a female 2 mm mm mm swim Slashes mm mm aw 7mm Elementary Statistics Looking at the Big Picture Example When Probabilities Can t Simply be dded El Background A statistics professor reports that the probability of a randomly chosen student in her class getting an A is PA025 and the probability of being a female is 1313060 El Response mmmmnm mm ammwsmms mmmmw mm C 2007 Nancy Pfenning De nition For some pairs of events if one occurs the other cannot and vice versa We can say they are non overlapping the same as disjoint or mutually exclusive 2mm mnwmm amnuwsmsm mm lm swim m 45 NonOverlapping Or Rule For any two nonoverlapping events A and B MAE BPAPB Note 1 Events female and getting anA d0 overlap Rule does not apply Note 2 The word or entails addition mmmmnm mm ammwsmms makmva heaiv mm 1114a Example Applying Non Overlapping Or Rule El Background Assuming adult male foot lengths have mean 11 and standard deviation 15 if we randomly sample 100 adult males the probability of their sample mean being less than 107 is 0025 probability of being greater than 113 is also 0025 El Question What is the probability of sample mean foot length being less than 107 or greater than 1 13 2mm mnwmm amnuwsmsm mm lm swim m 47 Example Applying Non Overlapping Or Rule Elementary Statistics Looking at the Big Picture El Background Assuming adult male foot lengths have mean 11 and standard deviation 15 if we randomly sample 100 adult males the probability of their sample mean being less than 107 is 0025 probability of being greater than 113 is also 0025 El Response mmmmnm mm ammwsmms makmva heaiv mm m 49 C 2007 Nancy Pfenning Example Intuiting Independent And Rule Example I ntuiting Independent And Rule El Background A balanced coin is tossed twice El Question What is the probability of both the rst and the second toss resulting in tails 2mm mnwmm amnuwsmgm mm tithe swim m 5 El Background A balanced coin is tossed twice El Response Looking Back Alternatively since there are 4 equally likely outcomes HH HT TH TT we know each has probability 14025 2mm mm mm ammw Statstics mm um aw mm m 52 Example When Probabilities Can t Simply be Multiplied Example When Probabilities Can t Simply be Multiplied El Background In a child s pocket are 2 quarters and 2 nickels He randomly picks a coin does not replace it and picks another El Question What is the probability of the first and the second coins both being quarters 2mm mnwmm amnuwsmgm mm tithe swim we El Background In a child s pocket are 2 quarters and 2 nickels He randomly picks a coin does not replace it and picks another El Response 2mm mm mm ammw Statstics mm um aw We L555 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning De nitions For some pairs of events whether or not one occurs impacts the probability of the other occurring and vice versa the events are said to be dependent If two events are independent whether or not one occurs has no effect on the probability of the other occurring 2mm mnwmm amnuwsmsm mm atthe whenquot m 56 Independent And Rule For any two independent events A and B PA BFNA PB Note The word and entails multiplication 2mm mm mm ammw Statstics mm um aw mm L1157 Sampling With or Without Replacement I Sampling with replacement is associated with events being independent I Sampling without replacement is associated with events being dependent 2mm mnwmm amnuwsmsm mm atthe whenquot in 5a Example Applying Independent And Rule Elementary Statistics Looking at the Big Picture El Background What is the probability of getting a female and then a male ifwe pick El Questions I 2 people With replacement from a household Where 3 of5 that is 06 are female I 2 people Without replacement from household Where 3 of 5 that is 06 are female I 2 people Without replacement from a large university Where 06 are female 2mm mm mm ammw Statstics mm um aw mm m 59 C 2007 Nancy Pfenning Example Applying Independent And Rule El Background What is the probability of getting a female and then a male ifwe pick El Responses I 2 people With replacement from a household Where 3 of5 that is 06 are fem e39 l 2 people Without replacement from household Where 3 of 5 that is 06 are female I 2 people Without replacement from a large university Where 06 are female 2mm mnwmm amnuwsmsm mm tithe swim in m Approximate Independence when Sampling Without Replacement Rule of Thumb When sampling Without replacement events are approximately independent if the population is at least 10 times the sample size Looking Ahead Because almost all reallife sampling is without replacement we need to check routinely if popukition is at least I On 2mm mm mm ammw Stalstics mm um aw We L562 Probability of Occurring At Least Once To nd the probability of occurring at least once we can apply the Not Rule to the probability of not occurring at all 2mm mnwmm amnuwsmsm mm tithe swim in as Example Probability of Occurring At Least Once Substitute Birthday Problem Elementary Statistics Looking at the Big Picture El BackgroundProbability of heads in coin toss is 05 El Question What is probability of at least one head in 10 tosses 2mm mm mm ammw Stalstics mm um aw We L564 C 2007 Nancy Pfenning Example Probability of OccurringAtLeast Lecture Summary in in Va a i i ies asic u es Once ngPbblt B Rl El BackgroundProbability of heads in coin toss is 05 El Probability de nitions and notation El Response PAlPnot AlPTTTTTTTTTT El Rules I Permissible probabilities I SumtoOne Rule N t Rul Looking Back Theoretically we could have used the Or I lt 0 1 ef 1 Rule adding the probabilities of all the possible ways to get I or Ru e or remover appmg events at least one heads However there are over 1000 ways 39 And R1113 for dependent evems 011089919 El Independence and sampling with replacement 0mm mwmm ammum mm mm BMW M mmme mm armamm WWW Mm L567 Elementary Statistics Looking at the Big Picture 13 Lecture 33 Two Categorical Variables More About ChiSquare tupotheses about Variables or Parameters uComputing Chisquare Statistic Details of Chisquare Test uConfounding Variables 0 2mm Nancy Ptenning Elementary Statistics Luuklng attne Big Picture C 2007 Nancy Pfenning E Looking Back Review El4 e 2mm Nancy Ptenning Stages of Statistics Data Production discussed in Lectures 14 Displaying and Summarizing Lectures 512 Probability discussed in Lectures 1320 Statistical Inference El 1 categorical discussed in Lectures 2123 El 1 quantitative discussed in Lectures 2427 El cat and quan paired 2sample severalsample Lectures 2831 El 2 categorical El 2 quantitative Elementary Statistics Luuklng attne Big Picture L33 2 H 0 and H a for 2 Categorical Variables El In terms of variables I E two categorical variables are related I H a two categorical variables are related El In terms of parameters I HO population proportions in response of interest are equal for various explanatory groups equal for various explanatory group C 2mm Nancy Pfenning Elementary Statistics Luuklng attne Big Picture I E population proportions in response of interest are Word not appears in Ho about variables Ha about parameters L33 3 Elementary Statistics Looking at the Big Picture Chi 0 mm Na Square Statistic Compute table of counts expected if H 0 true each is Column total x Row total expected Table total El Same as counts for which proportions in response categories are equal for various explanatory groups Compute chisquare test statistic X2 observed expected2 chisquare sum of expected ncy Ptenning Elementary Statistics Luuklng attne Big Picture L33 4 C 2007 Nancy Pfenning Observed and Expected Example 2 Categorical Variables 39 Data Expressions observed and expected commonly I Background Interested in relationship between used for chisquare hypothesis tests gender amp lenswear cc 9 contacts glasses none All More generally observed is our sample statistic female 121 32 129 282 expected is what happens on average in the 42 91 11352 4574 10000 population whenHO is true and there is no difference from claimed value or no relationship male 42 37 85 164 2561 2256 5183 10000 S All 163 69 214 446 El Question What do data show about relationship in the sample e 2337 Nancy Ptenning Eiernentaiy Statistics tanking attne Big Picture t33 5 e 2337 Nancy Ptenning Eiernentaiy Statistics tanking attne Big Picture t33 3 Example 2 Categorical Variables Data Example 2 Categorical Variables Test I Background Interested in relationship between I Background Interested in relationship between gender amp lenswear gender amp lenswear contacts glasses none All female 121 32 129 282 C G N Total 42911 11354 4574 100001 F 121 32 129 282 male 42 37 85 164 M 42 37 85 164 2561 2256 5183 10000 Total 163 69 214 446 All 163 69 214 446 El Response Females wear contacts more than males II Question Is there evidence of a relationship in the males wear glasses more larger population from which sample was taken proportions with none are close e 2337 Nancy Ptenning Eiernentaiy Statistics tanking attne Big Picture t33 3 e 2337 Nancy Ptenning Eiernentaiy Statistics tanking attne Big Picture t33 3 Elementary Statistics Looking at the Big Picture 2 C 2007 Nancy Pfenning i i E Example 2 Categorical Variables Test Example 2 Categorical Variables Test El Background Interested in relationship between El Background Interested in relationship between gender amp lenswear gender amp lenswear Expected Contacts Glasses None Total Femae 163282446103 69282446 44 214282446135 282 Observed Contacts Giasses None Totai EXPECled COHlaClS Glasses None Total Male 16316444660 691644462521416444679 164 446 Total 1 68 59 21 4 Total 69 214 446 El Response Compare observed and expected counts El Response First calculate expected counts 9different L33 in C 2mm Nancy Ptenning Eiernentaiy Statistics Lnnking attne Big Picture L33 i2 C 2mm Nancy Ptenning Eiernentaiy Statistics Lnnking attne Big Picture Example 2 Categorical Variables Test Example 2 Categorical Variables Test El Background Interested in relationship between El Background Interested in relationship between gender amp lenswear gender amp lenswear 121 103 31 32 44 33 129 135 03 121 103 3 1 32 44 33 i29 1352O3 103 39 44 39 135 39 103 39 44 39 135 39 42 602 37 252 85 792 42 602 37 252 85 792 54 58 2 05 54 58 05 60 25 79 6O 25 79 El Response Sum components to get chisquare El Response Next nd components standardized squared differences between observed and expected I 54 58 largest most impact from Is it large I 03 05 smallest least impact from Eiernentaiy Statistics Lnnking attne Big Picture L33 i6 L33 i4 C2uu7 Nancy Ptenning C 2mm Nancy Ptenning Eiernentaiy Statistics Lnnking attne Big Picture Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning ChiSquare Distribution Review Example ChiSquare Degrees of Freedom 2 observed expected chIsquare sum of follows Predlctable I Background Exammmg relat1onsh1p between pattern known as gender and lenswear chisquare distribution With df r l x c l I r number of rows possible explanatory values C G N Tom I 0 number of columns possible response values F 121 32 129 282 Properties of chisquare I Nonnegative based on squares M 42 37 85 164 I Meandf 1 for smallest 2x2 table Tom 163 69 214 446 Skewed right I Question How many degrees of freedom apply c 2mm Nancy Pfenning Elementary Staltsttcs Luuking althe Big Picture L33 17 c 2mm Nancy Pfenning Elementary Staltsttcs Luuklng althe Big Picture L33 18 Example ChiSquare Degrees of Freedom Chisquare Density curve I Background Examining relationship between For chisquare with 2 df PX2 2 6 005 gender and lensweari 9 If X2 is more than 6 Pvalue is less than 005 INote Degrees C G N Total of freedom tell h F 121 32 129 282 5 0w 3 M 42 37 85164varyfreely Total 163 69 214 446bef0re the rest are locked in area05 El Response row variable male or female has r column variable contacts glasses none has c 2395 5390 60 7395 039 chi square with 2 df dye 2 by3 table e 2mm Nancy Pfenning Elementary Statistics Luuking althe Big Picture L33 in e 2mm Nancy Pfenning Elementary Statistics Luuking althe Big Picture L33 21 Elementary Statistics Looking at the Big Picture 4 C 2007 Nancy Pfenning Example Assessing ChiSquare Example Assessing ChiSquare I Background In testing for relationship between I Background In testing for relationship between gender and lenswear in 2x3 table found X2 184 gender and lenswear in 2x3 table found X2 184 El Question Is there evidence of a relationship in El Response For df21x312 chisquare general between gender and lenswear not just in considered large if greater than 6 the sample 9186 large PValue small 9evidence of a relationship between gender and lenswear c 2mm Nancy Pfenning Eiernentary Statisties Luuking attne Big Pieture L33 22 c 2mm Nancy Pfenning Eiernentary Statisties Luuking attne Big Pieture L33 24 Example CheckingAssumplions Example CheckingAssumplions I Background We produced table of expected I Background We produced table of expected counts below right counts below right Observed Contacts Giasses None Total Expected Contacts Giasses None Total Observed Contacts Giasses None Totai EXPEClSd comacts masses None Total Female Male Total 163 69 214 446 II Question Are samples large enough to guarantee Response All expected counts are more than individual distributions approx normal so sum of 9 standardized components follows X distribution o 2mm Nancy Pfenning Eiernentary Statisties Luuking attne Big Picture L33 25 o 2mm Nancy Pfenning Eiernentary Statisties Luuking attne Big Picture L33 27 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning 1 E Example ChiSquare with Software Example ChiSquare with Software El Background Some subjects injected under arm Expected counts are printed below observed counts Decreased NotDecreased Total W1th Botox others W1th placebo After a month Botox 121 40 161 reported if sweating had decreased Output shown 8050 8050 Expected counts are printed below observed counts Placebo 40 121 161 Decreased NotDecreased Total 8050 8050 Botox 121 40 161 Total 161 161 322 8050 8050 ChiSq 20376 20376 placebo 40 121 161 20376 20376 31503 8050 8050 DF 1 PValue 02000 Total 161 151 322 El Response Sample s1zes large enough Proportions ChiSq 20376 20376 with reduced sweating 20376 20376 81503 v 2 D1 1 PValue oooo seem d1fferent P val 9d1ff s1gmf1cant 1 Question What do we conclude o 2007 Nancy Pfenning Conclude Botox reduces sweating o 2007 Nancy Pfenning Elementary Siaiisnes Luuking aiine Big Picture L33 23 Elementary Siaiisnes Looking aiine Big Picture L33 3n Guidelines for Use of ChiSquare Review Example Confounding Variables I Need random samples taken independently from 1 Background Students of all yearszx2 136 p 0000 tWO 01 more populations 1 1 On Campus 1 Off Campus 1 Total 1 Rate On Campus 1 1 Undecided 1 124 1 81 1 205 1 12420560 1 I Confoundmg varlables should be separated out1 1Decided 1 96 1 129 1 225 1 9622543 1 I Sample sizes must be large enough to offset non Underclassmen X2 0025 p 0873 IIOI Inality of distributions 1 Underclassmen 1 On Campus 1 Off Campus 1 Total 1 Rate On Campus 1 1 Undecided 1 117 1 55 1 172 1 11717268 1 I Need populatlons at least 10 times sample s1zes 1 Decided 1 82 1 37 1 119 1 8211969 1 Upperclassmen X2 1267 0262 1 Upperclassmen 1 On Campus 1 Off Campus 1 Total 1 Rate On Campus 1 1 Undecided 7 1 26 1 3 73321 1 1 Decided 1 14 1 92 1 106 1 141oe13 1 II Question Major dec and living situation related 0 2007 Nancy Pfenning o 2007 Nancy Pfenning Elementary Siaiisnes Luuking aiine Big Picture L33 31 Elementary Siaiisnes Luuking aiine Ellg Pieinre L33 32 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Confounding Variables Activity I Background II Complete table of total students of each gender on I Students of all years X2 136 p 0000 roster and count those attending and not attending I Underclassmen X2 02513 2 873 for each gender group Carry out a chisquare test I Upperclassmen X2 126710 2 262 to see if gender and attendance are related in El Response general I Students of all years 1700009evidence of relationship Attend NOt Attend TOtal 77 But 7 is confounding variable Fem ale Separate by suspected confounding variable M a e Underclassmen 17873 evidence of relationship 77 Total I Upperclassmen 172629evidence of relationship 7 c 2337 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L33 34 c 2337 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L33 33 Lecture Summary Inference for Cat9Cat More ChiSquare II Hypotheses about variables or parameters I Computing chisquare statistic I Observed and expected counts II Chisquare test I Calculations I Degrees of freedom I Chisquare density curve I Checking assumptions Testing With software I Confounding variables e 2337 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture Li 3 33 Elementary Statistics Looking at the Big Picture 7 C 2007 Nancy Pfenning 7 7 E l l l Looking Back Review Lecture 6 El 4 Stages of Statistics Quantitative Varables I Data Production discussed in Lectures 14 I Displaying and Summarizing Dlsplaysa Begln summarles El Single variables 1 cat Lecture 5 El Relationships between 2 variables I Probability I Statistical Inference llSummarize with Shape Center Spread llDispays Stemplots Histograms llFive Number Summary Outliers Boxplots C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg F39lCturE C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg F39lCturE LB 2 De nition Example Issues to Consider El Distribution tells all possible Values of a I Background Intro stat student earnings year before variable and how frequently they occur game Earmleg ar OS Brittany 3 Dominique 7 Adam 1 I Questions I Data representative of What population I Responses unbiased I How to summarize I Sample average 3776 9 population average lt 5000 C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg F39lCturE LB 3 C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg F39lCturE LB 4 Elementary Statistics Looking at the Big Picture 1 C 2007 Nancy Pfenning Example Issues to Consider De nitions U BaCRgl Ollndi 111th Stat Student earnings Summarize values of a quantitative variable by i Eamfg telling shape center spread B tt 3 ngijriihue 7 III Shape tells which values tend to be more or imam less common ii Responses El Center measure of what is typical in the I Data represent all students at that univeti f dlstrlbutlon of a quantltatlve varlable g Back I Responses unbiased 1f I How to summarize Task at hand D Spr adz measure Of how muCh the l Sample average 3776 9 population average lt 5000 dIStrlbutlon s values vary lLaaking Ahead This is an inference question I 0 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture LB 5 0 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture LB E Definitions Displays of a Quantitative Variable El Symmetric distribution balanced on either side of Displays help see the shape of the distribution center El Stemplot El Skewed distribution unbalanced lopsided Advantage most detail I Skewed left has a few relatively low Values I Disadvantage impractical for large data sets El Skewed right has a few relatively high values U HiStOgl am II Outliers values noticeably far from the rest 39 Afivantage works We f r any 5126 data set D Unimodal Singlepeaked ll3 Dislacivantage some detail lost El X El Bimodal twopeaked 0 p 0 U f o 11 1 11 H h I Advantage shows outliers makes comparisons C9Q D In 0rm a V21 ues equa y 09111111011 at S ape I Disadvantage much detail lost I Normal a particular symmetric bellshape 0 2mm Nancy Prenning Eiernentary Statistics Luuking attne Big Picture LB 7 c ZEIEI7 Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture LB 8 Elementary Statistics Looking at the Big Picture 2 C 2007 Nancy Pfenning De nition Example Constructing a Stemplot D Stemplot vertical list of Stems each El Background Masses in 1000 kg of 20 dinosaurs 5 0000010204060707101111121517171829325056 fouowed by honzomal list Of one39dlg leaves El Question Display with stemplot What does it tell Stems 139d1glt leaves us about the shape gt gt gt 0107 WWW Mum mm m mm m WWW mm amumm WWW m Wu Example Constructing a Stemplot Modi cations to Stemplots El Background Masses in 1000 kg of 20 dinosaurs D Toofew Stems Split 00 0001 02 04 06 0707 10 11 11 12 15 171718 29 32 50 56 El Response DO not Skip the 4 stem Why I Split in 2 1st stem gets leaves 04 2quot gets 59 I Split in 5 1st stem gets leaves 01 2quot gets 23 etc st th Long tang skewed l Split in 10 1 gets 0 10 gets 9 I 1 peak El Too many stems Truncate last d1g1ts Most below 2000 kg a few unusually heavy mmmmnm mm amnmvsmms makmva heaiv mm swim mm mm tithe aw 7mm 2 mm mm mm Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Splitting Stems El Background Credits taken by 14 other students 47111111131314141517171718 Questions El I What shape would We guess for other nontraditional studenw I How to construct stemplot to make shape clear 2mm mnwmm amnuwsmsm mm tithe swim m is Example Splitting Stems El Background Credits taken by 14 other students 47111111131314141517171718 El Responses I Expect shape skeWed due to I Stemplot lst attempt has too few stems 0 i 4 7 11111334457778 2mm mm mm ammw mm mm um aw mm mu Example Truncating Digits El Background 1 nutes spent on computer day before 0 10 20 30 30 30 30 45 45 60 60 60 67 90 100 120 200 240 300 420 El Question How to construct stemplot to make shape 0 ear 2mm mnwmm amnuwsmsm mm tithe swim L521 Example Truncating Digits Elementary Statistics Looking at the Big Picture El Background 1 nutes spent on computer day before 010 20 30 30 30 30 45 45 60 60 60 67 90 100 120 200 240 300 420 El Response Stems 0 to 42 too many truncate last digit work with 100 s stems and 10 s leaves Skewed 39 most times less than 100 mimtes but a few had umsually long times 2mm mm mm ammw mm mm um aw mm m2 C 2007 Nancy Pfenning Displays of a Quantitative Variable De nition El Stemplot El Histogram El Boxplot 2mm mnwmm amnuwsmsm mm atthe swim 1524 El Histogram to display quantitative values Divide range of data into intervals of equal width u Find count or percent or proportion in each Use horizontal axis for range of data values vertical axis for countpercentproportion in each 2mm mm mm ammw Statstics mm um aw mm L525 Example Constructing a Histogram Example Constructing aHistogram El Background Prices of 12 used upright pianos 100 450 500 650 695 1100 1200 1200 1600 2100 2200 2300 El Question Construct a histogram for the data What does it tell us about the shape 2mm mnwmm amnuwsmsm mm atthe swim ma El Background Prices of 12 used upright pianos 100 450 650 695 1100 1200 1200 1600 2100 2200 2300 El Response We opted to put 500 as left endpoint of2nd interval be consistent apriee of1000 would go in 3rd interval not 2nd 2mm mm mm ammw Statstics mm um aw mm Lazs Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning De nitions Review El Shape tells which values tend to be more or less common El Center measure of what is typical in the distribution of a quantitative variable El Spread measure of how much the distribution s values vary 2mm mnwmm amnuwsmsm mm atthe swim Lam De nitions El Median a measure of center I the middle for odd number of values I average of middle two for even number of values El Qualtiles measures of spread I 151 Quartile Ql has onefourth of data values at or below it middle of smaller half I 3ml Quartile Q3 has threefourths of data values at or below it middle of larger halt By hand for odd number of values omit median to nd quartiles 2mm mm mm ammw Statstics mm um aw mm Lacm De nitions El Percentile value at or below which a given percentage of a distribution s values fall A Closer Look Q1 is 25th percentile Q3 is 75 h percentile El Range difference between maximum and minimum values El Interqualtile range tells spread of middle half of data values written IQRQ3Ql 2mm mnwmm amnuwsmsm mm atthe swim L531 Ways to Measure Center and Spread Elementary Statistics Looking at the Big Picture El Five Number Summary 1 lVLinimum 2 Q1 3 Median 4 Q3 5 Maximum El Mean and Standard Deviation more useful but less straightforward to nd WWW Mm mmwmm mammal We L532 C 2007 Nancy Pfenning Example Finding 5 Number Summary and I QR Example Finding 5 Number Summary and I QR El Background Credits taken by 14 nontraditional El Background Credits taken by 14 nontraditional students 4 71111 11 131314141517171718 students 4 71111 11 131314141517171718 El Question What are the Five Number Summary El Response range and IQR 1 Minimum 77 2 Q1 3 Median 7 4 Q3 77 5 Maximum 77 Range isi IQR is i CJZEIEI7 Nancy Pfenning Elementary Statistics Looking atthe Big Picture LB 33 CJZEIEI7 Nancy Pfenning Elementary Statistics Looking atthe Big Picture LB 35 De nition Displays of a Quantitative Variable The 15 TimesIQR Rule identifies outliers II Stemplot II below Qll5IQR considered low outlier II Histogram II above Q3l 5IQR considered high outlier El Boxplot 15TimesIQR Rule to Identify Outliers IQRsQS Q1 15 IQR 15 IQR l l l l I r I or 63 low outliers high outliers CJZEIEI7 Nancy Pfenning Elementary Statistics Looking atthe Big Picture LB 36 CJZEIEI7 Nancy Pfenning Elementary Statistics Looking atthe Big Picture LB 37 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning i J E De nition Example Constructing Boxplot El Background Credits taken by 14 nontraditional students had 5 No Summary 4111351718 El Questions I Are there outliers I How do we construct a boxplot A boxplot displays median quartiles and extreme values with special treatment for outliers 1 Bottom whisker to minimum nonoutlier 2 Bottom of box at Q1 3 Line through box at median 4 Top of box at Q3 5 Top whisker to maximum nonoutlier Outliers denoted C 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture LE 38 e 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture LE 39 Exam le C onstructz39n Box at Exam le C onstructz39n Box at p g P p g P El Background Credits taken by 14 nontraditional El Background Credits taken by 14 nontraditional students had 5 No Summary 4 11 135 17 18 students had 5 No Summary 4 11 135 17 18 D D CVEditS about I IQR between I I and I 7 shape is leftskewed Maximum189 19 f 15xIQR7 Q3179 39 9 I Q 1155IIQI Ilcgt71110ut111ers Median1359 14 I t 52 Q Q 7777 1g ou iers 77 21119 E IQRQSQ1 9 15mm 15 iQR l 39 39 I 61 63 Minimum 49 4 low outiiers high outiiers c 2mm Nancy Pfenning Eiernentary Statistics Luuking atthe Big Picture i c ZEIEI7 Nancy Pfenning Eiernentary Statistics Luuking atthe Big Picture LB 43 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Lecture Summary Quantitative Displays Begin Summaries El Display stemplot histogram Shape Symmetric or skewed Unimodal Normal Center and Spread I median and range IQR u identify outliers EIEI a display with boxplot 2 mm mm mm aimuw mm mm mm aw 7mm Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning 7 E l l l Looking Back Review Lecture 23 El 4 Stages of Statistics I Data Production discussed in Lectures 14 Inference for Categorical Variable I Displaying and Summarizing Lectures 542 More About Hypothesis Tests I Probability discussed in Lectures 1320 I Statistical Inference Examples of Tests With 3 Forms of Alternative El 1 categorical con dence intervals hypothesis tests EIHOW Form of Alternative Affects Test 1 quantitative ElWhen P Value is Small Statistical Significance ElHypothesis Tests in LongRun ElReIating Test Results to Confidence Interval categorical and quantitative 2 categorical 2 quantitative e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture L23 2 l l 39 7 Three Types of Inference Problem Review Hypothesis Test About p Review In a sample of 446 students 055 ate breakfast State null and alternative hypotheses H o and H a 1 What is our best guess for the proportion of all Null is status quo alternative rocks the boat students who eat breakfast p gt 100 Hoppo VS Ha pltpo P01ntEst1mate p 72 pO 2 What interval should contain the proportion of 1 l Consider sampling and study designl all Students Who eat breakfaSt 2 Summarize with standardize to Z assuming Confidence Interval that H0 p 2 390 is true consider if Z is large 3 Do more than half 50 of all students eat 3 Find Pvalueprobof Z this far abovebelowaway breakfast from 0 consider if it is small Hypothesis Test 4 Based on size of Pvalue choose H 0 or H a e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture L23 3 e 2mm Nancy Pfennan Elementary etatlstles Leeklng attne ale F39leture L23 4 Elementary Statistics Looking at the Big Picture 1 Checking Sample Size Cl vs Test ll Confidence Interval Require observed counts in and out of category of interest to be at least 10 ma 2 X 2 10 n1 13n X2 10 III Hypothesis Test Require expected counts in and out of category of interest to be at least 10 assume p p0 TWO 2 10 n1 190 Z 10 e 2mm Nancy F39fErlrllrlg Elementary Statisties Luuklng attne Eilg F39lcture L23 5 C 2007 Nancy Pfenning E Example Checking Sample Size in Test I Background 304000075 students picked 7 at random from 1 to 20 Want to test H 0 p005 vs Ha pgt005 II Question Is n large enough to justify finding Pvalue based on normal probabilities e 2mm Nancy F39fErlrllrlg Elementary Statisties Luuklng attne Eilg F39lcture L23 6 Example Checking Sample Size in Test I Background 304000075 students picked 7 at random from 1 to 20 Want to testHO p005 vs Ha pgt005 il Response n P0 nlpo Looking Back For con dence interval checked 30 and 370 both at least 10 e 2mm Nancy F39fErlrllrlg Elementary Statisties Luuklng attne Eilg F39lcture L23 8 Elementary Statistics Looking at the Big Picture Example T est with gt Alternative Review CI Note Step 1 requires 3 checks I Is sample unbiased Sample proportion has mean 005 I Is population 210n Formula for sd correct I Are npo and nlpo both at least 10 Find or estimate Pvalue based on normal probabilities 1 Students are typical h n39 04 1 issue at hand 2 pr005 sd of is 005l 005nd Z 075 quot oo51 o05 39 400 3 Pvalue PZ Z 229 is small just over 001 4 RejectHo conclude Ha picks were biased for 7 e 2mm Nancy F39fErlrllrlg Elementary Statisties Luuklng attne Eilg F39lcture L23 9 C 2007 Nancy Pfenning Example Test with Less Than Alternative Example Test with Less Than Alternative I Background 111230 of surveyed commuters at a El Background 111230 of surveyed commuters at a university walked to school uanerSlty walked to SCh001 II Question Do fewer than half of the university s D Response FlrSt Wnte H03 VS Ha commuters walk to school 1 Students need to be representative in terms of year 2 Output9 13 2 Test and CI for One Proportion Test and CI for One Proportion Test ofp05 vsplt05 Test ofpO5 vsplt05 Sample X N Sample p 9507 Upper Bound ZValue P Value Sample X N Sample p 9507 Upper Bound ZValue PValue 1 111 230 0482609 0536805 053 0299 1 111 230 0482609 0536805 053 0299 3 Pvalue 4 RejectHo c 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 in c 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 i3 Example Test with Less Than Alternative Example Test with Not Equal Alternative CI Note Pvalue is a lefttailed probability because I Background 43 of Florida s community college alternative was less than students are disadvantaged II Question Is disadvantaged at Florida Keys Community College 169356475 unusual Test and CI for One Proportion Test of p 043 vs p not 043 Sample X N Sample p 950 CI Z Value P Value 1 169 356 0474719 0422847 0526592 170 0088 o 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 M o 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L23 i5 Elementary Statistics Looking at the Big Picture 3 C 2007 Nancy Pfenning Example Test with Not Equal Alternative Example Test with Not Equal Alternative 393 BaCkgmund 43 0f Florida s community COllege CI Note Pvalue is a twotailed probability because students are disadvantaged alternative was not equal El Response First write H 0 vs H a 1 356043 3561043 both210 pop210356 2 p z Test and CI for Cine Proportion Test of p 043 vs p not 043 Sample X N Sample p 9507 CI Z Value P Value 1 169 856 0474719 0422847 0526592 170 0088 3 Pvalue 4 RejectHO C 2mm Nancy Ptenning Eiementary Statistics Luuking atthe Big Picture L23 i7 e mi Nancy Ptenning Eiementary Statistics Luuking atthe Big Picture L23 iB E 9 90959899 Rule Outside Probabilities Onesided or Twosided Alternative I Form of alternative hypothesis impacts 05 Pvalue I Pvalue is the deciding factor in test e area025 1 area0 l area 005 area025 I Alternative should be based on what researchers hopefear suspect is true 3005 before snooping at the data area01 i i i 4545 I 70 just gm 1 6459 L1 645 I Z I If lt or gt is not obv1ous use twoSided 4960 l I 2325 1 329326 alternative more conservatlve 72576 2576 e mi Nancy Ptenning Eiementary Statistics Luuking atthe Big Picture L23 in e 2007 Nancy Ptenning Eiementary Statistics Luuking atthe Big Picture L23 2i Elementary Statistics Looking at the Big Picture 4 C 2007 Nancy Pfenning E Example How F arm of Alternative A ects Test Example How F arm of Alternative A ects Test I Background 43 of Florida s community college El Background 43 of Florida s community college students are disadvantaged students are disadvantaged II Question Is disadvantaged at Florida Keys 539 Response NOW Wme H03 VS H03 169356475 unusually high 1 Same checks of data production as before 2 Same 0475z170 Test of p 043 vs p gt 043 Sample X N Sample p 9501 Lower Bound Z Value PValue 1 169 356 0474719 0431186 170 0044 3 Now Pvalue 4 Reject H 0 e 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture L23 23 e 2mm Nancy Ptenning Eiernentary Statistics Looking attne Big Picture L23 25 i 39 e Pvalue for One or TwoSided Alternative Thinking About Data I Pvalue for onesided alternative is half Before getting caught up in details of test Pvalue for twosided alternative consider evidence at hand I Pvalue for twosided alternative is twice Pvalue for onesided alternative For this reason twosided alternative is more conservative larger Pvalue harder to reject Ho e 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture L23 2B e 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture L23 27 Elementary Statistics Looking at the Big Picture 5 C 2007 Nancy Pfenning Example Thinking A bout Data at Hand El Background 43 of Florida s community college students are disadvantaged At Florida Keys the rate is 475 Question Is the rate at Florida Keys signi cantly lower El 2mm mnwmm Eiemenuwsuusucs mm tithe swim 1232s Example Thinking About Data at Hand El Background 43 of Florida s community college students are disadvantaged At Florida Keys the rate is 475 El Response 2mm mm mm ammw Stalslics mm um aw mm m cm De nition alpha 01 cutoff level which signi es a Pvalue is small enough to reject H 0 Eiemenuwsuusucs mm tithe swim 12331 How Small is a Small PValue Elementary Statistics Looking at the Big Picture I Avoid blind adherence to cutoff 05005 I Take into account 1 Past considerations is 10 Written in stone or easily subject to debate Future considerations What would be the consequences of either type of error I Rejecting H0 even though it s true I Failing to reject He even though it s false I Consider decisions encountered so far U 2mm mm mm ammw Stalslics mm um aw mm m 32 C 2007 Nancy Pfenning Example Reviewing P values and Conclusions El Background Consider our prototypical examples I Are random number selections biased PvaluF001 l I Do fewer than half of commuters walk Pvalue4299 I Is disadvantaged signi cantly different PvaluF0088 I Is disadvantaged signi cantly higher Pvalue0044 El Question What conclusions did we draw based on those Pvalues 2mm mnwmm amnuwsmsm mm mm aiwmme 1233 Example Reviewing P values and Conclusions El Background Consider our prototypical examples I Are random number selections biased Pvalue4011 I Do fewer than half of commuters walk Pvalue0299 I Is disadvantaged signi cantly different Pvalue4088 I Is disadvantaged signi cantly higher Pvalue0044 El Response Consistent with 005 as cutoff Oi I P value001 l 9Rej ect 7 I P value02999 Reject 7 I P value0088 9Rej ect 7 I P value0044 9Rej ect 7 2mm mm mm gummy Statstics mm um aw mm m 35 Example CutO s for Small quotP Value El Background Bookstore chain will open new store in a city if there s evidence that its proportion of college grads is higher than 026 the national rate El Question Choose cutoff 010 005 001 I if no other info is provided I if chain is enjoying considerable pro ts owners are eager to pursue new ventures if chain is in financial difficulties can t afford losses if unsuccessful due to too few grads 2mm mnwmm amnuwsmsm mm mm aiwmme mas Example CutO s for Small quotP Value Elementary Statistics Looking at the Big Picture El Response Choose cut0ff010 005 001 if no other info is provided El use77 I if chain is enjoying considerable pro ts owners are eager to pursue new ventures El use 7 I if chain is in financial difficulties can t afford loss if unsuccessful due to too few grads El use 2mm mm mm gummy Statstics mm um aw mm m as C 2007 Nancy Pfenning De nition Role of Sample Size n Statistically significant data produce Pvalue small enough to rejectHo Z plays a role l Large 11 may reject H 0 even though observed proportion isn t very far frompo Z 13 290 2 from a practical standpoint iPo1Po po1P0 TL Reject Ho ifPvalue small if Z large if I Sample proportion 13 far from p0 Very small Pvalue strong evidence against Ho but p not necessarily very far from po l Small 11 may fail to reject H 0 even though I Sample size n large it is false I Standard deviation small if pois close to 0 or 1 Failing to reject false H0 is 2 type of error e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 3a e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 4n i l De nition Hypothesis Test and LongRun Behavior I Type I Error reject null hypothesis even Repeatedly carry out hypothesis tests of p05 though it is true false positive based on 20 coinflips using cutoff 5 i Probability is cutoff Ct In the long run 5 of the tests will reject I Type 11 Error fail to reject null HO p05 even though it s true hypothesis even though it s false false negative e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 M e 2mm Nancy F39fErlrllrlg Elementary Statistles Leeklng attne Big F39lcture L23 42 Elementary Statistics Looking at the Big Picture 8 C 2007 Nancy Pfenning J i J HypotheSIS Test and LongRun Behav10r Confidence Interval and Hypothesis Test Results 20 mi lps test H0 pgggy sb gggggg equal 50 l Con dence Interval range of plausible values TlTITHTH39lTHHT e HH 39 p39 p i eads45 Zquot45quotquotVal e 655 A I Hypothesis Test decides if a value is plausible HTI HHTHHTITHTHTlTHHT i Proportion of head 40 2389 p39Vame39371 4 IIIfOImally Z iagvaalueaaw 4 El If 170 is in confidence interval don t re ect Ho 7170 39 El pr0 is out51de confidence interval reject Ho 7170 THHHl tTHHHTHT HHH Z 2 24pvaiue 025 Relationship between 95 confidence interval pr p lhead 3975 l and twosided test with 05 as cutoff for pvalue 0 llips oi 20 o 95 chests do not reject Ho If 0 IS here I39e GCt HO 15 39 5 oi tests reject Ho i i l Cl 95 confidence interval i for population proportion V V Tl39H HTTTHTTHHTHHH proportion of heads 8204O 2289 pevalue37t A do not reiect Ho C 2mm Nancy Pfenning Elementary Statistics Looking atthe Big Picture L23 43 C 2mm Nancy Pfenning I If W is here do not relem Ho ppo I L23 44 Example Test Results Based on C Example Test Results Based on C I Background A 95 confidence interval for I Background A 95 confidence interval for proportion of all students choosing 7 at proportion of all students choosing 7 at random from numbers 1 to 20 is random from numbers 1 to 20 is 0055 0095 0055 0095 I Question Would we expect a hypothesis test I Response to reject the claim p005 in favor of the claim pgt005 Elementary Statistics Looking at the Big Picture 9 Example CI Results Based on Test El Background A hypothesis test did not reject HO p0 5 in favor of the alternative H 11 plt05 El Question Do we expect 05 to be contained in a con dence interval for p 2mm mnwmm amnuwsmgm mm tithe swim mm C 2007 Nancy Pfenning Example CI Results Based on Test El Background A hypothesis test did not reject HO p05 in favor of the alternative Ha plt05 El Response 2mm mm mm ammw Statstics mm um aw mm m 5 Lecture Summary iiI ore Hypothesis Tests for Proportions El Examples with 3 forms of alternative hypothesis El Form of alternative hypothesis I Effect on test results I When data render formal test unnecessary I Pvalue for lsided vs 2sided alternative Cutoff for small Pvalue Statistical signi cance role of n Type I or H Error Hypothesis tests in longrun EIEIEIEI Relating tests and confidence intervals 2mm mnwmm amnuwsmgm mm tithe swim Liam Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning E l l 9 Looking Back Review Lecture 17 III 4 Stages of Statistics Continuous Random Variables I Data Production discussed in Lectures 14 I Displaying and Summarizing Lectures 512 Normal Distribution I Probability El Finding Probabilities discussed in Lectures 1314 El Random Variables introduced in Lecture 15 Relevance of Normal Distribution EIContinuous Random Variables I Binomial discussed in Lecture 16 689599 Rule for Normal RVs EIStandardizingUnstandardizing Probabilities for StandardNonstandard Normal RVs D samphng Disnilbunons I Statistical Inference e 2mm Nancy Pfenning Elementary Statistics Lnnking attne Big Picture e 2mm Nancy Pfenning Elementary Statistics Luuking attne Big Picture L17 2 Role of Normal Distribution in Inference Discrete vs Continuous Distributions I Goal Perform inference about unknown I Binomial Count X population proportion based on sample El discrete distinct possible values like numbers proportion 1 2 3 I Strategy Determine behavior of sample I Sample Proportion I3 proportion in random samples with known El also discrete distinct values like count poPUIation Proportion I Normal Approx to Sample Proportion I Key Result Sample proportion follows El continuous follows normal curve normal curve for large enough samples E Mean p standard deviation W Looking Ahead Similar approach will be taken with means it Elementary Statistics Looking at the Big Picture 1 C 2007 Nancy Pfenning Jr 1 E Sample Proportions Approx Normal Review Example Variable Types I Proportion of tails in n16 coin ips p05 has El Background Variables in survey excerpt A 050 2 MW 2 0125 shape approx normal l age breakfast comp credits I Proportion of lefties p0l in n100 people has no 120 15 no 120 16 M 0170 W 003 shape approx normal 1908 yes 40 14 W 5 n100 El Question Identify type catdiscquan contquan I Age I Breakfast Probanllrly mommy quot a 25 Sn if I 5 39lquot la is I Comp daily time in min on computer imam e prsampis ereeemen le rhanded Elli I Credits c 2mm Naney F39fErlrllrlg Elementary etatlstles Luuklng attne Big Pletere Ll7 5 c 2mm Nancy F39fErlrllrlg Elementary etatlstles Luuklng attne Big Pletere Ll7 6 Example Variable Types Probability Histogram for Discrete RV I Background Variables in survey excerpt HiStOgram for male Shoe SiZCX represents probability by area of bars i age breakfast comp credits 1967 no 120 15 I pX S 9on left 2008 no 120 16 I on ri ht 1908 yes 40 14 A PX lt 9 g ll Response I Age I Breakfast in H r snee male I C d 391 t39 39 39 t omp a1 y me In mm on compu er For discrete RV strict inequality or not matters I Credits C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Big F39lcture Ll 7 a C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Big F39lcture Ll 7 a Elementary Statistics Looking at the Big Picture 2 C 2007 Nancy Pfenning Definition Density Curve for Continuous RV Density curve smooth curve showing prob dist of Density curve for male foot lengthX represents continuous RV Area under curve shows prob PTObablhtY by area under CUIVe that RV takes value in given interval Looking Ahead Most commonly used density curve is normal z but to perform inference we also use t F and chisquare curves Probability of X less than 9 Probability of X less than or equal to 9 m i z D2 a l M J M s 7 la 9 in n la is l4 l5 la o2 N m r XFootlenglhimale mi Kl It PX g 9 PX lt 9 N i 7 J A F l 2 a Continuous RV strict inequality or not doesn t matter c 2mm Naney F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg Pleture Ll7 in c 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg Pleture Ll7 ll 6895997 Rule for Normal Data Review 6895997 Rule Normal Random Variable Values of a normal data set have Sample at random from normal population for El 68 within 1 standard deviation of mean sampled value X a RV probability is El 95 within 2 standard deviations of mean El 68 thatX is within 1 standard deviation of mean I 997 within 3 standard deviations of mean El 95 thatX is within 2 standard deviations of mean 3895997 Rule for Normal Distributions ll 997thatX1s Within 3 standard dev1ations of mean area16 area16 I 58 oi valuesigt areaaozs an 7 areal025 a oi 5 A a L areaTg 5 957a ofvalues e 0 I l l V rw meansad Illealll392sd mainijal gxiueietnnsa mealnzsd me a ntasd quot7950 72 r A 3 3920 MI30 c 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg Pleture Li7 l2 c ZEIEI7 Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg Pleture U7 is Elementary Statistics Looking at the Big Picture 3 C 2007 Nancy Pfenning 6895997 Rule for Normal RV Example 689599 7 Rule for Normal R V Looking Back We use Greek letters to denote 1 Background IQ for randomly chosen adult population mean and standard deviation is normal RV Xwith H 100 a 15 mean Itquot Standard deV39at39On a El Question What does Rule tell us about distribution of X area16 area16 I gt H684 area025 area025 are 0015 area 15 95 H a 9197 1 i i M 30 p 20 M 0 1 0 n2a M3a 02uu7 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L17 14 02uu7 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L17 15 Example 689599 7 Rule for Normal R V Example Finding Probabilities with Rule I Background IQ for randomly chosen adult I Background IQ for randomly chosen adult is normal RVXwith u 100 0 15 is normal RVXwith u 100 0 15 III Response We can sketch distribution of X I Question Prob of IQ between 70 and 130 areai16 area16 area16 A area16 lt gt lt gt area39025 4 M gt area025 area39025 4 3929 area025 510015 95 arear0g5 are DO 95 area S 0 I 93997 igl 0 E I 997 74M I 55 70 5 100 1i15 1 30 1115 10 02uu7 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L17 17 02uu7 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L17 1E Elementary Statistics Looking at the Big Picture 4 C 2007 Nancy Pfenning i J Example Finding Probabilities with Rule Example Finding Probabilities with Rule I Background IQ for randomly chosen adult I Background IQ for randomly chosen adult is normal RVXwith u 100 0 15 is normal RVXwith u 100 0 15 III Response Prob of IQ bet 70 and 130 III Question Prob of IQ less than 70 area16 area16 area16 area16 lt gt lt gt area025 4 M gt area025 area025 4 39m gt area025 are 0075 area 15 are 10015 area 15 95 95 H 0 997 7 Tog 0 2 997 7A 55 7390 5 100 1l15 1l30 1115 55 7390 5 loo 1l15 1l30 11115 IQ IQ 02uu7 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L17 2n 02uu7 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L17 21 Example Finding Probabilities with Rule Example Finding Probabilities with Rule I Background IQ for randomly chosen adult I Background IQ for randomly chosen adult is normal RVXwith u 100 0 15 is normal RVXwith u 100 0 15 III Response Prob of IQ less than 70 I Question Prob of IQ less than 100 area16 area16 area16 A area16 lt gt lt gt area025 4 M gt area025 area025 4 3929 area025 510015 95 arear0g5 are DOiS 95 area S 0 93997 74 0 E 997 7 55 7390 5 100 1l15 1 30 1115 55 7390 5 loo 1l15 1 30 1115 IO 10 02uu7 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L17 23 02uu7 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L17 24 Elementary Statistics Looking at the Big Picture 5 C 2007 Nancy Pfenning 1 J Example Finding Probabilities with Rule Example Finding Values of X with Rule I Background IQ for randomly chosen adult I Background IQ for randomly chosen adult is is normal RVXwith u 100 0 15 normal RV Xwith u 100 0 15 III Response II QuestionProb is 0997 that IQ is between area16 area16 areai16 area16 lt gt lt gt area025 4 M gt area025 area025 4 39m gt area025 are 0075 area 15 are 10015 area 15 o 7 3 I d o 7 2 7 g 5395 7390 5 13900 1i15 1i30 115 5395 7390 5 lIOO 1i15 1i30 11175 IQ IQ 02uu7 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L17 2B 02uu7 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L17 27 Example Finding Values of X with Rule Example Finding Values of X with Rule I Background IQ for randomly chosen adult is I Background IQ for randomly chosen adult normal RVXwith M 100 0 15 is normal RVXwith M 100 0 15 III Response Prob 0997 that IQ bet and III Question Prob is 0025 that IQ is above areai16 area16 area16 A area16 lt gt lt gt area025 4 M gt area025 area025 4 3929 area025 are 0015 area 15 are 0015 area 15 o 0 2 fax a 0 2 f9 70 5395 7390 5 13900 1i15 1 30 1115 5 5 7390 5 lIOO 1i15 1 30 1115 IO 10 02uu7 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L17 2a 02uu7 Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L17 an Elementary Statistics Looking at the Big Picture 6 C 2007 Nancy Pfenning VJ Example Finding Values of X with Rule Example Using Rule to Evaluate Probabilities El Background IQ for randomly chosen adult is normal RV Xwith H 2 100 a 15 El Background Foot length of randomly chosen adult male is normal RVXwith M 11 U 15 in El Response Prob 1s 0025 that IQ 1s above 1 Question How unusual is foot less than 65 inches area16 area16 area 16 area 16 68 area025 area 15 95 area025 area025 area025 areg0015 95 areag 15 areg0015 0 i I 997 i I 0 1 997 r gt1 55 70 5 100 115 130 145 is Bill 5 1 1 13925 139410 1 55 lQ Xmale foot length C 2mm Nancy Pfenning Elementary Statistics Looking atthe Big Picture L17 32 C 2mm Nancy Pfenning Elementary Statistics Looking atthe Big Picture L14 33 Example Using Rule to Evaluate Probabilities Example Using Rule to Estimate Probabilities El Background Foot length of randomly chosen adult male is normal RVXwith u 11 a 15 in El Response Footlt65 El Background Foot length of randomly chosen adult male is normal RVXwith u 11 a 15 in El Question How unusual is foot more than 13 inches area16 area16 area16 area16 1 gt lt gt area1025 68 area025 areazlozs 4 Fa area025 39 are 0015 area 15 are 0015 area 15 a 95 g 3 95 g9 l 1 9197 l 1 l 1 49197 l 1 65 80 5 11 125 140 155 65 80 5 11 125 140 155 Xmale loot length Xmale loot length 0 2mm Nancy Pfenning Elementary Statistics Looking atthe Big Picture L14 35 0 2mm Nancy Pfenning Elementary Statistics Looking atthe Big Picture L14 3B Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning i E De nition Review Example Using Rule to Estimate Probabilities iI z score or standardized value tells how many standard deviations below or above the mean the original value is I Background Foot length of randomly chosen adult male is normal RVXwith a 11 a 15 in a Response PXgtl3 value mean A Z m r area16 3 D Notation for Population Z U LL AC2 39 area025 I zgt0 for x above mean 1 95 emery I zlt0 for x below mean 1997 r R F El Unstandardize c l 20 65 50 5 11 25 13940 1 55 M Xmale foot length C 2mm Nancy Pfenning Eiementary Statistics Luuking attne Big Picture LN 38 C 2mm Nancy Pfenning Eiementary Statistics Luuking attne Big Picture Li 7 39 Standardizing Values of Normal RVs Example Standardized Value of Normal R V Standardizing to Z lets us avoid sketching a different 393 BaCRgl Ollndi Typical nightly hours 51610th conege curve for every normal problem we can always Students 110111131 M 7i 0 15 refer to same standard normal 2 curve I Question How many standard deviations below or above mean is 9 hrs area16 area 6 i i gt 68 area025 area025 are 0015 area 15 0 7 997 ye i T I I i 3 2 1 0 1 2 3 Z c ZEIEI7 Nancy Pfenning Eiernentary Statistics Luuking attne Big Picture Li7 4i Elementary Statistics Looking at the Big Picture 8 r Example Standardized Value of Normal R V I Background Typical nightly hours slept by college students normal it 7 a 15 El Response Standardize to Z 9 is standard deviations mean n 5 hoursx 0 9 standardized Nu C 2mm Nancy F39fErlrllrlg Elementary Statlstlcs Luuklng attne Eilg F39lCture Ll 7 43 J C 2007 Nancy Pfenning Example S tandardizin g Unstandardizin g Normal R V I Background Typical nightly hours slept by college students normal it 7 a 15 I Questions I What is standardized value for sleep time 45 hours I If standardized sleep time is 25 how many hours is it C 2mm Nancy F39fErlrllrlg Elementary Statlstlcs Luuklng attne Eilg F39lCture Ll4 44 Example S tandardizin g Unstandardizin g Normal R V I Background Typical nightly hours slept by college students normal u 7 a 15 CI Responses I What is standardized sleep time for 45 hours I If standardized sleep time is 25 how many hours is it C 2mm Nancy F39fErlrllrlg Elementary Statlstlcs Luuklng attne Eilg F39lCture Ll4 4e Elementary Statistics Looking at the Big Picture Interpreting zscores Review This table classifies ranges of zscores informally in terms of being unusual or not Size of z Unusual z greater than 3 extremely unusual 2 between 2 and 3 very unusual 2 between 175 and 2 unusual 2 between 15 and 175 maybe unusual depends on circumstances z between 1 and 15 somewhat lowhigh but not unusual 2 less than 1 quite common Looking Ahead Inference conclusions will hinge on whether or not a standardized score can be considered unusual C 2mm Nancy F39fErlrllrlg Elementary Statlstlcs Luuklng attne Eilg F39lCture Ll 7 47 C 2007 Nancy Pfenning Example Characterizing Normal Values Based Example Characterizing Normal Values Based on z Scores 0n z Scores I Background Typical nightly hours slept by college El Background Typical nightly hours slept by college students normal u 7 a 15 students normal u 7 0 15 El Questions El Responses I How unusual is a sleep time of45 hours z167 I Sleep time of45 hours I How unusual is a sleep time of 1075 hours z25 I Sleep time of 1075 hours Size of z Unusual z greater than 3 extremely unusual z between 2 and 3 very unusual 2 between 175 and 2 unusual 2 between 15 and 175 maybe unusual depends on circumstances 2 between 1 and 15 somewhat lowhigh but not unusual 2 less than 1 quite common c2uu7 Nancy Pfenning Eiememaiy Statistics Luuking attne Big Picture MAME c2uu7 Nancy Pfenning Eiememaiy Statistics Luuking attne Big Picture Lizi an N ormal Probability Problems Example Estimating Probability Given 2 I I Estimate probability given 2 I I Background Sketch of 6895997 Rule for Z El Probability close to 0 or 1 for extreme Z I Estimate 2 given probability area16 A area16 l Estimate probability given nonstandard x area025 3968 l Estimate nonstandard x given probability a39ea39 25 0015 15 a 95 area lg 0 i 997 gt1 i i T I 3 2 71 0 1 2 3 Z I Question Estimate PZlt147 C 2mm Nancy Pfenning Eiementary Statistics Looking attne Big Picture Li 7 5i C 2mm Nancy Pfenning Eiementary Statistics Looking attne Big Picture Ln 52 Elementary Statistics Looking at the Big Picture 10 C 2007 Nancy Pfenning Example Estimating Probability Given 2 Example Estimating Probability Given 2 11 Background Sketch of 6895997 Rule for Z 11 Background Sketch of 6895997 Rule for Z area16 1 area16 area16 area 68 68 gt area025 area39025 area025 are 0015 ar a 15 x 95 l 95 a e 709 0 s QIEW l gt1 0 1 227 i i l 73 2 0 1 2 393 Z a 72 71 0 1 2 la 1 47 Z 11 Response PZlt1 47 11 Question Estimate PZgt075 0 2mm Nancy Pfenmng E1ememary 5mm Lunkwg atthe E119 F39mture L17 54 0 2mm Nancy Pfenmng E1ememary 5mm Lunkwg atthe E119 F39mture L17 55 Example Estimating Probability Given 2 Example Estimating Probability Given 2 11 Background Sketch of 6895997 Rule for Z areao50 area7 areai16 11 Background Sketch of 6895997 Rule for Z area16 area16 b r area025 EBBH area 025 39r g are OO15 area 15 4 35 I h 0 3995 gt1 l n amalgam 1 997 7 1 71 IAVIIIIIIIIIIIIIIIIrA l t I l t 393 2 r1 0 1 2 1393 Z 3 2 1 0 1 2 3 75 V r 2 11 Response PZgt075 I III Question Estimate PZlt28 C 2mm Nancy Pfenmng E1ememary 5mm Lunkwg atthe E19 F39mture L17 57 C m7 Nancy Pfenmng E1ememary 5mm Lunkwg atthe E19 P1cture L17 58 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Estimating Probability Given 2 o V 1i r E Normal Probability Problems l Estimate probability given 2 El IProbability close to 0 or 1 for extreme Z I Estimate 2 given probability l Estimate probability given nonstandard x I Estimate nonstandard x given probability e ZEIEI7 Nancy Prenning Eierneritaiy Statistics Luuking atthe Big Picture U761 0 3VR Vq l 3 2 1 o 1 2 3 Z 28 I Response PZlt2 8 c 2667 Nancy Pfenning Eiernentary Statistics Luuking atthe Big Picture Li 7 EU I Example Probabilities for Extreme 2 I Background Sketch of 6895997 Rule for Z area16 area16 I 39 gt Z I Question Estimate PZlt145 e ZEIEI7 Nancy Prenning Eiementary Statistics Luuking atthe Big Picture 768 area025 area025 are 0015 magg 5 lt1 95 l o 997 7 TT i i T I l 39 3 72 c1 0 1 2 3 L17 EZ Example Probabilities for Extreme 2 I Background Sketch of 6895997 Rule for Z area16 A area16 E 7684 area025 area025 are t0015 area 5 95 gt g o 997 e h i i T i i i 3 72 c1 0 1 2 3 Z I Response PZltl45 e ZEIEI7 Nancy Prenning Eiementary Statistics Luuking atthe Big Picture w 64 Elementary Statistics Looking at the Big Picture Example Probabilities for Extreme 2 I Background Sketch of 6895997 Rule for Z area16 area16 lt I gt 68 area025 area025 are 0015 area 15 41 95 l g o 997 7 h 1 7 T 39 I 39 3 2 1 0 1 2 3 Z I Question Estimate PZgt38 C 2997 Nancy Pfenmng Etementary Staustms Luukmg atthe Ehg F39mture L17 as C 2007 Nancy Pfenning Example Probabilities for Extreme 2 I Background Sketch of 6895997 Rule for Z area16 area16 4 I 39 gt 68 area025 area025 are 0015 area 15 95 gt g 0 997 1 7 T 39 7T 3 2 1 0 1 2 3 Z I Response PZgt38 C 2997 Nancy Pfenmng Etementary Staustms Luukmg atthe Ehg F39mture L17 B7 Example Probabilities for Extreme 2 I Background Sketch of 6895997 Rule for Z area16 area16 I 39 gt 768 area025 area025 are 0015 area 5 lt1 95 l g o 997 7 TT 1 1 T I i 39 3 2 1 0 1 2 3 Z I Question Estimate PZlt13 C 2997 Nancy Pfenmng Etementary Staustms Luukmg atthe Ehg F39mture L17 BE Elementary Statistics Looking at the Big Picture Example Probabilities for Extreme 2 I Background Sketch of 6895997 Rule for Z area16 area16 4 E 7684 area025 area025 are 0015 area 5 95 gt g o 997 71 1 T 1 3 2 1 0 1 2 3 Z I Response PZltl3 C 2997 Nancy Pfenmng Etementary Staustms Luukmg atthe Ehg Pcmre L17 7n C 2007 Nancy Pfenning Example Probabilities for Extreme 2 Example Probabilities for Extreme z I Background Sketch of 6895997 Rule for Z I Background Sketch of 6895997 Rule for Z area16 areai16 area16 area16 lt gt lt gt area39025 3968 area025 area39025 3968 area025 are 0015 area 15 are 0015 area 15 0 1 1 37 7 a g 0 I237 5 gt I 7 I I 71 I T I I I t I T I I 3 2 71 0 1 2 3 3 2 71 0 1 2 3 Z Z I Question Estimate PZgt235 II Response PZgt235 cnum Nancy F39fErIrIIng EIememary StatIstIcs LuukIng althe Etg F39Icture L17 71 cnum Nancy F39fErIrIIng EIememary StatIstIcs Luukmg althe Etg F39Icture L17 73 N ormal Probability Problems Example Estimating 2 Given Probability I Estimate probability given z I Background Sketch of 6895997 Rule for Z 1 Probability close to 0 or 1 for extreme Z I I Estimate 2 given probability I area16 A area16 I Estimate probability given non standarcl x area025 3968 areaquot 025 l Estimate nonstandard x given probability quot i0015 a 95 area 5 0 I 997 gt1 1 I T I 3 2 1 0 1 2 3 Z I Question Prob is 001 that Zltwhat value cnum Nancy Pfenmng EIememary StatIstIcs LuukIng althe Etg F39Icture L17 74 02uu7 Nancy F39fErIrIIng EIememary StatIstIcs LuukIng althe Etg F39Icture L17 75 Elementary Statistics Looking at the Big Picture 14 C 2007 Nancy Pfenning Example Estimating 2 Given Probability Example Estimating 2 Given Probability I Background Sketch of 6895997 Rule for Z I Background Sketch of 6895997 Rule for Z area025 area16 area16 4 I 39 gt 68 larea3901 area39025 68 area025 agaaDOi 5 95 gt 91001 5 95 away 5 0 997 gt o 39 97 l I T I I Z i I gr I i 7T 393 272 0 1 2 3 3 72 71 2 1 2 3 I Response Prob is 001 that Zlt II Question Prob is 015 that Z gtwhat value cnum Nancy Pfenning Elementary Statistics Luuking atthe Big Picture LI7 77 cnum Nancy Pfenning Elementary Statistics Luuking atthe Big Picture LI7 78 Example Estimating 2 Given Probability Normal Probability Problems I Background Sketch of 6895997 Rule for Z l Estimate probability given Z area16 El Probability close to 0 or 1 for extreme z area15 I Estimate 2 given probability l Estimate probability given nonstandard x I l Estimate nonstandard x given probability rI n G YIAVIIllIIIIIlllt r 2 0 1 2 393 Z H I 3 II ResponsezProb is 015 that Zgt C 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture LI 7 an C m7 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture LI 7 Bi Elementary Statistics Looking at the Big Picture 15 C 2007 Nancy Pfenning Example Estimating Probability Given x Example Estimating Probability Given x El Background Hrs sleth normal t 7 a 15 El Background Hrs sleth normal M 7 a 15 area16 area16 area16 39 gt 1 I area 684 area025 area025 are 0015 15 95 l area rug 4 O l 23997 7 i i 0 Ir I VII1111157529 3 gt2 71 0 1 2 3 3 V2 2 El Question Estimate PXgt9 D Response CJZEIEI7 Nancy F39fErrrrrrrg Eiementary Statistics Luukrng atthe Big Picture L17 EZ CJZEIEI7 Nancy F39fErrrrrrrg Eiementary Statistics Luukrng atthe Big Picture L17 84 Example Estimating Probability Given x Example Estimating Probability Given x a Background Hrs slethnonnal i 7 or 15 U Background Hrs slethnonnal a 7 o 15 area68 area area16 area16 area025 area025 510015 3995 area g 5 O I i E f 0 i iH 19 l 3 72 1 1 2 3 3 2 I67 0 67 1 2 3 2 El Question Estimate P6ltXlt8 D Response WDWNWWEWQ Semen5mm WW am 3 9 mm W ES I A Closer Look 0 67 a71a39 0 67 are the Miles of the z 0W Elementary Statistics Looking at the Big Picture 16 C 2007 Nancy Pfenning Normal Probability Problems Example Estimating x Given Probability l Estimate probability given 2 El Background Hrs sleth normal 1 7 a 1 5 El Probability close to 0 or 1 for extreme Z 16 I Estimate 2 given probability L l Estimate probability given nonstandard x areawzs 68 Eek025 I Estimate nonstandard x given probability 3 0015 95 areaTilms 0 I i 997 7 gt 3 2 1 0 1 2 is Z ll Question 004 is PXlt 32007 Naney F39fErlrllrlg Elementary Statistles Luuklng attne Elg Pletere Ll7 ea 32007 Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Elg Pletere Ll7 89 Example Estimating x Given Probability Example Estimating x Given Probability I Background Hrs sleth normal H 7 a 15 El Background Hrs sleth normal 3911 7 a 15 area16 area04 area16 area16 4 RC2 r 08 area025 Ml area025 are 0015 area 15 o 1337 7 gtl em 0 a 95 gtl g I 5 l l 2 i i 997 l i 77 gt3 2 e 1 2 3 3 2 1 0 1 2 3 2 El Response Z ll Question 020 is PXgt C 2007 Nancy F39fErlrllrlg Elementary Statistles Leenng attne Eilg F39lcture Lli Bl C 2007 Nancy F39fErlrllrlg Elementary Statistles Leenng attne Eilg F39lcture Li 7 92 Elementary Statistics Looking at the Big Picture Example Estimating x Given Probability C 2007 Nancy Pfenning area025 I Background Hrs sleth normal 7 area 20 area16 area1 I i 015 gt area025 El Response C ZEIEI7 Nancy Pfenning Eiementaiy Statistics Luuking aims Big Picture are 0015 area 15 95 I i Qg i i i i 7 3 2 1 0 1 2 3 2 mi 94 E Strategies for Normal Probability Problems l Estimate probability given nonstandard x El Standardize to Z El Estimate probability using Rule l Estimate nonstandard x given probability El Estimate Z El Unstandardize to x C ZEIEI7 Nancy Pfenning Eiementaiy Statistics Looking aims Big Picture Li 7 as Lecture Summary Normal Random Variables Relevance of normal distribution 6895997 Rule for normal RVs Standardizingunstandardizing Probability problems I Find probability given 2 EIEIEIEIEI I Find 2 given probability I Find probability given x I Find x given probability C ZEIEI7 Nancy Pfenning Eiementaiy Statistics Luuking aims Big Picture Continuous random variables density curves LMBB Elementary Statistics Looking at the Big Picture Lecture 3 Designing Studies Focus on Experiments uDefinitions uRandomization uControl uBIind Experiment nPitfalls uSoecific 39 Desiqns 2 mm mm mm swim shims mm We aw 7mm C 2007 Nancy Pfenning Looking Back Review El 4 Stages of Statistics I Data Production I Displaying and Summarizing I Probability I Statistical Inference mmmmnw mm Eiementaivs tstics mmmmw mm m2 Looking Back Review El 2 Types of Study Design I Observational study record variables values as they naturally occur El Drawback confounding variables due to self assignment to explanatory V ues El Example Men who drink beer are more prone to lung cancer than those who drink red wine what is the confounding variable here I Experiment researchers control values of varia El If well designed provides more convincing evidence of causation 2mm mnwmm amnuwsmsm mm We gimme m De nitions Elementary Statistics Looking at the Big Picture El Factor an explanatory variable in an experiment El Treatment value of explanatory variable imposed by researchers in an experiment A control group individuals receiving no treatment or baseline treatment may be included for comparison If individuals are human we call them subjects 2mm mm mm ammw Statstics mm mm aw mm m C 2007 Nancy Pfenning Example Randomized controlled experiment and ot ers to 1 Population assignments 2 mm mm mm Eiemenhw shims mm tithe aw 7mm El Background To test if sugar causes hyperactivity researchers assign some children to low h39h evels of sugar consumption Sample El Question What is the advantage of random Example Randomized controlled experiment El Background To test if sugar causes hyperactivity researchers randomly assign some children to low and others to high levels of sugar consumption El Response mumm mm Eiementaivstatstics makinva heaiv mm m Experiment vs Observational Study Populatio n a 2mm mm Eiemenhw shims mm tithe aw 7mm In an experiment researchers decide who has low sugar intake L and who has high H Sample Sugar intake as not yet been determined Researchers assign sugar intake L or H Experiment vs Observational Study In observational study individuals have already chosen low L or high H sugar intake Population Sample Researchers make no changes to sugar intake mumm mm Eiementaivstatstics makinva heaiv We L12 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Randomization at rst or second stage Consider two selection issues in our sugar hyperactivity experiment El What individuals are included in the study El Who consumes low and high amounts of sugar At which stage is randomization important 2mm mnwmm amnuwsmsm mm tithe swim a u Randomization at First or Second Stage Individuals studied may be volunteers Otherwise noncompliance may be an issue Population Sample Assignment to sugar L or H must be random Volunteering which treatment to get is not OK mmmmnm mm ammmysmm makmva heaiv mm L111 Must an experiment have a control group Recall our de nition El Experiment researchers manipulate explanatory variable observe response Thus experiment may have no control group El if all subjects must be treated El if simulated treatment is risky El if the experiment is poorly designed As long as researchers have taken control of the explanatory variable it is an experiment 2mm mnwmm amnuwsmsm mm tithe swim L112 Elementary Statistics Looking at the Big Picture De nitions Three meanings of control El We control for a confounding variable in an observational study by separating it out El Researchers control who gets what treatment in an experiment by making the assignment themselves ideally at random El The control group in an experiment consists of individuals who do not receive a treatment per se or who are assigned a baseline value of the explanatory variable mmmmnm mm ammmysmm inakmvanheaiv mm mm C 2007 Nancy Pfenning Doubleblind experiments Two pitfalls may prevent us from drawing a conclusion of causation when results of an experiment show a relationship between the socalled explanatory and response variables El If subjects are aware of treatment assignment El If researchers are aware of treatment assignment 2 mm mm mm swim shims mm tithe aw 7mm De nitions El The placebo effect is when subjects respond to the idea of treatment not the treatment itself El A placebo is a dummy treatment El A blind subject is unaware of which treatment heshe is receiving El The experimenter effect is biased assessment of or attempt to in uence response due to knowledge of treatment assignment El A blind experimenter is unaware of which treatment a subject has received mmmmnm mm armaments makmva heaiv mm Example Subjects not blind El Background Suppose after children are randomly assigned to consume either low or high amounts of sugar researchers nd proportion hyperactive is greater for those who consumed higher amounts El Question Can we conclude sugar causes hyperactivity swim shims mm tithe aw 7mm Elementary Statistics Looking at the Big Picture Example Subjects not blind El Background Suppose after children are randomly assigned to consume either low or high amounts of sugar researchers nd proportion hyperactive is greater for those who consumed higher amounts El Response El Improvement mmmmnm mm armaments makmva heaiv mm C 2007 Nancy Pfenning 3 1 Example Experimenters not blind Example Experimenters not blind El Background Suppose after children are randomly El Background Suppose after children arerandomly assigned to diets sweetened either arti cially or with ass1g ed t0 dletS sweetened miller a l CIauY 0 W1th sugar researchers nd proportion hyperactive rs sugar researchers find proportion hyperactive is greater for those who consumed sugar El Response greater for those who consumed sugar I Question Can we conclude sugar causes hyperactivity More problematic if responses are assessed Improvement c 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 i9 c 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 2i 2 2 Best evidence of causation Other pitfalls in experimentation In general conclusions of causation are most I Lack of realism lack of ecological validity COHVlnClng 1f a relatlonShlp has been I Hawthorne effect subjects behave differently established in a randomized controlled because of awareness of participation in doubleblind experiment experiment A Closer Look In the original studies reporting a relationship D Noncomphance between sugar and Imperaetivity Conducted in the 1970 I Treatments unethical experimenters may have been airvare thlie children 39s diet when they assessed bEhGVlOI randomized controlledsingleblineb U Treatments Impracticalimposs1ble t0 Impose Lilam studies since then havefailed to establish a relationship c 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 22 C 2mm Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture L3 23 Elementary Statistics Looking at the Big Picture 5 C 2007 Nancy Pfenning Example Hawthorne e eet lack of realism Example Hawthorne e eet lack of realism El Background Suppose researchers want to determine if TV makes people snack more I While study participants are presumably wailing to be interviewed half are assigned to a room with a TV on and snacks the other half to a room with no TV and snacks See if those in the room with TV consume more snacks El Question If participants in the room with TV snack more can we conclude that in general people snack more when they watch TV 2mm mnwmm amnuwsmsm mm tithe swim ma El Background Suppose researchers want to determine if TV makes people snack more I While study participants are presumably waiting to be interviewed half are assigned to a room with a T Von and snacks the other half to a room with no TV and snacks See if those in the room with T Vconsume more snacks El Response if people suspect they re observed TV and snacking habits different in contrived setting 2mm mm mm ammw Statstics mm um aw We ma Example Non compliance in experiment Example Non compliance in experiment El Background To test if sugar causes hyperactivity researchers randomly assign 50 children to low and 50 to high levels of sugar consumption 20 drop out of each group For remaining children 30 in each group suppose proportion hyperactive is substantially greater in the highsugar group El Question Can we conclude sugar causes hyperactivity 2mm mnwmm amnuwsmsm mm tithe swim L127 El Background To test if sugar causes hyperactivity researchers randomly assign 50 children to low and 50 to high levels of sugar consumption 20 drop out of each group For remaining children 30 in each group suppose proportion hyperactive is substantially greater in the highsugar group El Res onse 7 makes treatment and control groups different in ways that may affect response 2mm mm mm ammw Statstics mm um aw mm mm Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Another awed experiment El Background To test if stuttering is a learned rather than inborn trait a researcher in Iowa in 1939 randomly assigned subjects to I Control 11 orphans in ordinary speech therapy I Treatment 11 orphans badgered and interrupted in sessions with speech therapist Of the 11 in treatment group 8 became stutterers Question What s wrong with this experiment El 2mm mnwmm amnuwsmgm mm mm swim mu Example Another awed experiment El Background To test if stuttering is a learned rather than inborn trait a researcher in Iowa in 1939 randomly assigned subjects to I Control 11 orphans in ordinary speech therapy I Treatment 11 orphans badgered and interrupted in sessions with speech therapist Of the 11 in treatment group 8 became stutterers El Response 2mm mm mm ammw Statstics mm um aw mm L132 Examples Treatments impossibleimpractical El men get married sooner promoted quicker and earn higher wages El There is a link betweenand low SOC 00110111 C status 111 women Height is impossible to control Weight is dif cult to control Socioeconomic status is too costly to control 2mm mnwmm amnuwsmgm mm mm swim in Elementary Statistics Looking at the Big Picture Modi cations to randomized experiment El Blocking Divide rst into groups of individuals who are similar with respect to an outside variable that may be important in relationship studied El Paired design Randomly assign one of each pair to receive treatment the other control Before and a er is a common paired design Looking Back block71g M to experimentation as Strati cation is to sampling 2mm mm mm ammw Statstics mm um aw mm L134 C 2007 Nancy Pfenning Example Blocked experiment Example Blocked experiment El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Question How to incorporate blocking if researcheis suspect location plays a role in relationship between type of sunscreen and amount of time spent in sun 2mm mnwmm amnuwsmgm mm tithe swim ms El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Response 2mm mm mm ammw Statstics mm um aw mm L137 Example Paired experiment Example Paired experiment El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Question How to incorporate paired design if researchers suspect location plays a role in relationship between type of sunscreen and amount of time spent in sun 2mm mnwmm amnuwsmgm mm tithe swim ma El Background Study tested theory that use of stronger sunscreen causes more time in sun Before vacation 40 students given weak sunscreen 40 given strong Students recorded time spent in sun each day El Response 2mm mm mm ammw Statstics mm um aw mm mu Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Paired experiment The paired design helps to ensure that treatment and control groups are as similar as possible in all other respects so that if their responses dijfkr we have evidence that the treatment is responsible 2mm mnwmm amnuwsmms mm tithe swim L141 Example Combining Paired and Two Sample Designs El Background Studies often randomly assign one group to a placebo and the other to a drug Responses to the variable of interest are assessed before and after a period of time then compared to see benefits or side effects El Question What aspect of the design is twosample and What aspect is paired 2mm mm mm ammw Stalstics mm um aw mm m 44 Example Combining Paired and Two Sample Designs El Background Studies often randomly assign one group to a placebo and the other to a drug Responses to the variable of interest are assessed before and after a period of time then compared to see benefits or side effects El Response twosample paired 2mm mnwmm amnuwsmms mm tithe swim m as Elementary Statistics Looking at the Big Picture Lecture Summary Experiments De nitions Randomization 2 stages ofselection Control group Blind study design I Subjects blind to avoid placebo elfect DUDE I Researchers blind to avoid experimenter elfect Other pitfalls of experiments lack of realism Hawthorne effect noncompliance unethical or impractical treatment B U Speci c experimental designs I Blocked Paired ortwosample 2mm mm mm ammw Stalstics mm um aw We L147 C 2007 Nancy Pfenning Looking Back Review Lecture El 4 Stages of Statistics I Data Production discussed in Lectures 14 Categorlcal amp Quantltatlve Varlable Displaying and Summarizing Lectures 512 Inference in SeveralSample Design Probability discussed in Lectures 1320 I Statistical Inference a uCompare and Contrast Several and 2sample 1wegmald s ss d quotL 39 r s 3923 a lquantitative discussed in Lectures 24 27 uVarIatIon Among Means or V thn Groups D m and qua palred 25mm uF Statistic as Ratio of Variation D 2 categori a1 u 2 quantitative Role of Sample Size Inference Methods for C9Q Review Display amp Summary Several Samples Review l Paired reduces to lsamplel El Display Side by side boxplots El Focused on mean of differences I One boxplot for each categorical group I TwoSample 2sample 2 similar to lsample I 39 All Share Same quantimiVe 50319 El Focused on difference between means I SeveralSample need new distribution F I FiVe Number Smma es 100kin at boxplots El Focus on difference among means 39 Means and Standard De da ons Looking Ahead Inference r population relationship nuses on means and standard deviations El Summarize Compare 2mm mnwmm amnuwsmsm mm mm awaits mus 2mm mm mm gummy Statstics mm um aw mm m 4 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Notation Two vs SeveralSample Inference Sizes Means I Similar test statistic standardizes difference Sample I no of groups compared n1n7ni sum to N 51322 151 overall 5 among sample means taking sample Sizes and Popmat39on I 1 2 quot standard deviations into account I Different severalsample test statistic F focuses on El Squared differences of means in numerator El Squared standard deviations variances in denominator Procedure called ANOVA ANalysis Of VAriance c 2mm Nancy Pfenning Elementary Statistics Luuking aime Big Picture Lari 5 c 2mm Nancy Pfenning Elementary Statistics Luuking aime Big Picture Lari E Two vs SeveralSample Inference l and F Distributions I Similar test statistic standardizes difference I Left sampled 100 values from at distribution among sample 11163115 takmg sample SlZeS and I Right squared the 100 values from t distribution Standard devlauons into account 2 Squaring makes F nonnegative rightskewed For 2 groups of equal Sizes and 01 039 2 F t 25 7 so and conclusions including Pvalue are the same 20 40 Frequency oi 8 51 l l i Frequency A m o o o e i 0 0 a 2 ii ii g 0123456159 Sampled 100 values from t distribution Squared sample 01 100 values from t distribution LED 8 e mi Nancy Pfenning Elementary Statistics Luuking althe Big Picture Lam 7 e ZEIEI7 Nancy Pfenning Elementary Statistics Luuking althe Big Picture Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Two vs SeveralSample Statistics What Makes 1 or F Statistics Large I How different are sample means I Large diff among sample means in numerator I How spread out are the distributions I Small spreads in denominator I How large are the samples I Large sample sizes denominator of denominator t Elif20 32 82 12 n1 n2 n1a ci 2 02 H2632 W men W I 1 n2n2 ntM1 1 721 1s n2 1gts n1 ls ltN 1 tnl 1 n2 1153 m 1s l I c 2mm Naney Pfenning Elementary Statistles Luuklng attne Big Picture LSD ll c 2mm Nancy Pfenning Elementary Statistles Luuklng attne Big Picture LSD l2 Example Sample SDs E ect 0n PValue Example Sample SDs E ect 0n PValue I Background Boxplots with 9 01 2 3 E2 4 i3 5 El Background Boxplots withn cl 3 52 2 4 33 5 could appear as on left or right depending on sds could appear as on left or right depending on sds 6 ntext sample 6 Context sample 57 mean monthly pay 5 a mean monthly pay 47 El 39 in 1000s for 3 4 El 39 in 1000s for 3 37 B racialethnic 3 E3 racialethnic 27 groups 2 groups El Question For which scenario does the difference El Response Difference between means appears among means appear more signi cant more signi cant on smaller sds overlap c 2mm Nancy Pfenning Elementary Statistles Luuklng attne Big Picture LED is c 2mm Nancy Pfenning Elementary Statistles Luuklng attne Big F39lcture LED is Elementary Statistics Looking at the Big Picture 3 C 2007 Nancy Pfenning E Example Sample SDs E ect on Conclusion Example Sample SDs E ect on Conclusion El Background Boxplots with E1 3 22 4 933 5 El Background Boxplots Withil 3 532 4 533 5 could appear as on left or right depending on sds COUld appear as 011 left or ght dependmg 011 SdS 6 Context sample 6 Context sample 5 a mean monthly pay 5 mean monthly pay 4 E 39 in 1000s for 3 4 E 39 in 1000s for 3 3 E3 racialethnic 3 E racialethnic 2 groups 2 groups I Questlon For which scenario are We more likely D Response Scenario on 2 smaller S 1 s 9 to reject hypothes1s of equal population means larger F Statistic9smaller PValue9 likelier to reject H o conclude c 2mm Nancy Pfenning Eiememary Statistics Luuking atthe Big Picture Lam i6 c 2mm Nancy Pfenning Eiememary Statistics Luuking atthe Big Picture Lam i8 i i 39 7 Measuring Variation Among and Within Numera tor of F Difference Among Means I 711031 9 02 H2652 32 7115quot 32 1 1 of Squared diffs among Groups iltn1 1gtse n2 1gts m we ltN Igt ssa 53 42 54 42 55 42 I Numerator variation among groups 3 Of Freedom for Groups I How different are 2721 EI from one another I Denominator variation within groups 3 diffs among Groups I How spread out are samples sds 517 31 monthly earnings in 1000sfor 3 racialethnic groups n 1 i e mi Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture Lam 18 e ZEIEI7 Nancy Pfenning L a a i I Lam 2n Elementary Statistics Looking at the Big Picture 4 C 2007 Nancy Pfenning E Numerator of F Difference Among Means Denominator of F Spread Within Groups X Note numerator of F is the same for both scenarios i SSE Su a of Squared Error within Groups because the difference among means is the same SSE 511 2511l582511582 a El DFE Degrees f Freedom for Error s DFEV I 15 3 4 E a g II MSE Mean Square rror w1th1n Groups 2 SSE If MSEDFE12 quot1 5 1 Z 3 monthly earnings quot2 5 f2 I 4 in 1000s for 3 n2 2 5 3 5 15 E Z 4 metaletth groups fl 1 39 1 c2uu7 Nancy Pfenning Elementary Statisties Luuking attne Big Pietuie Lari 2i c2uu7 Nancy Pfenning L g g I Lari 22 i i 39 7 Denominator of F Spread Within Groups The F Statistic II Note denominator ofF is smaller for the F mo 1 52 71252 m2 n1T m2 I 1 scenario on the right because of less spread 711 18g n2 308 I I 305 N I M SG i 2 Is2 large 47 E MSE 25 3 E3 measures difference among sample means relative to spreads and sample sizes IfFis large reject H0 M1 M2 M3 onc ue population means differ II Because the numerators are the same F the quotient is considerably larger on the right e 2mm Nancy Pfenning Elementary Statisties Luuking attne Big Picture tau 23 e 2mm Nancy Pfenning Elementary Statisties Luuking attne Big Picture tau 24 Elementary Statistics Looking at the Big Picture 5 C 2007 Nancy Pfenning Example Size of Standardized Statistics El Background Say standardized statistic is 2 El Question Is 2 large I For 2 I For t I For F 7 2mm mmmnm amnuwsmsm mm mm swim mm Example Size of Standardized Statistics El Background Say standardized statistic is 2 El Response I 22 combined tail probs 4 I F2 large I F2 large depends on based on total sample sizeN and number of groups 2mm mm mm ammw Statstics mm um aw mm m 27 F and its Degrees of Freedom Family of F curves all nonneg rightskewed Spreads vary depending on DFG I l in numerator DFE N I in denominator 2mm mmmnm amnuwsmsm mm mm swim muzs Example Degrees of Freedom for F Elementary Statistics Looking at the Big Picture El Background Consider these F distributions I F with I5 N390 F with DFG2 DFE12 written F212 El Questions I What are degrees of freedom ifI5 N390 I What are I andN ifDFG2 DFE12 2mm mm mm ammw Statstics mm um aw mm m 29 C 2007 Nancy Pfenning E Example Degrees of Freedom for F Example Assessing Size of F Statistic I Background Consider these F distributions El BaCRgl Ollndi Say F3 for DFG4 DFE385 08 7 i FWlth 15 N390 m e 552 3223 3 i3 2i12 333i I FWith DFG2 DFE12 0 5 t 05 7 III Responses 04 7 03 7 I 02 7 gt Since 1 121 Since N1N 312 N 7 7 00 I 1 2 3 F El Questions Is F3 large Will we reject a claim that the 5 population means are equal e 2mm Nancy Pfenning Elementary Statisties Laaking attne Big Picture tau 3i e 2mm Nancy Pfenning Elementary Statisties Laaking attne Big Picture tau 32 i i 39 7 Example Assessing Size of F Statistic Example Assessing F for Different DF I Background F3 for DFG4 DFE385 I Background Say F3 for DFG2 DFE12 F distribution for 4 dt in numerator 1 0 7 385 df in denominator 15 N390 F distribution for 2 df in numerator 12 dt in denominator l3 N15 PF gt 3018 PFgt30878 00 J I i 0 i i i i i i i i 1 2 3 F 1 2 3 A 5 6 7 8 El Responses PVal001859Very little area past F3 El Questions Is F3 large 9 F IS ReJeCt 0131111 that the 5 POPUIatlon What would we conclude if F2 for DFG2 means are equal DFE12 e 2mm Nancy Pfenning Elementary Statisties Laaking attne Big Picture tau 34 e 2mm Nancy Pfenning Elementary Statisties Laaking attne Big Picture tau 35 Elementary Statistics Looking at the Big Picture 7 C 2007 Nancy Pfenning E Example Assessing F for Different DF The F Statistic El Background F23 for DFGZZ DFEZlZ F n1il 302 n2 2 2 n ff 2 I 1 721 1sn2 1s nI 1siltN I F distribution for 2 of in numerator 12 df in denominator l3 N15 MSG i 2 Is 2 large for DFG2 DFE12 05 7 MSE 25 NO gt 2 I 1 measures difference among sample means relative to spreads and sample s1zes IfFis large reject H0 ul 2 a2 3 El Responses Pval008789F3 is PVal for F 2 must be 9Reject H o 9Conclude populatingg ieags ay be equal cum Nancy Pfenning Ei onc ue population means differ C 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture Lam 38 2 Example Drawing Conclusions Based on F Example Drawing Conclusions Based on F I Background Earnings for 5 sampled individuals I Background Earnings for 5 sampled individuals from three racialethnic groups had means 3 4 5 from three racialethnic groups had means 3 4 5 in thousands of dollars ANOVA procedure in thousands of dollars ANOVA procedure resulted in F 2 which in this case is not large resulted in F 2 which in this case is not large I Question What do we conclude about mean I Response Since F is not large sample means earnings for populations in the three racialethnic differ significantly from one another groups Conclude population mean earnings 31337 Nancy WEWVWQ E EWENIEN Statistics LDDWQ atthE Big F39iEWE L3H 39 31337 Nancy WEWVWQ E EWENIEN Statistics LDDWQ atthE Big F39iEWE L3H 4i Elementary Statistics Looking at the Big Picture 8 Example Role ofn in ANO VA Test ii Background Earnings for 12 instead of 5 sampled individuals from three racialethnic groups had means 3 4 5 in thousands of dollars ANOVA procedure resulted in F 48 and a Pvalue of 0015 C 2007 Nancy Pfenning E Example Role ofn in ANO VA Test ii Background Earnings for 12 instead of 5 sampled individuals from three racialethnic groups had means 3 4 5 in thousands of dollars ANOVA procedure resulted in F 48 and a Pvalue of 0015 ii Response Conclude population mean earnings for the three groups samples help provide more evidence against Ho e ZEIEI7 Nancy Pfenning Eiementary Statistics Luuking attne Big Picture LED 44 ii Question What do we conclude about mean earnings for populations in the three racialethnic groups c ZEIEI7 Nancy Pfenning Eiementary Statistics Luuking attne Big Pietuie LED 42 E 7 1 Mean of F Since thas sd typical distance of values from 0 approximately 1 and F is similar to squaringt distribution mean of F is approximately 1 F distribution for 4 df in numerator 385 at in denominator 5 N390 F distribution for 2 df in numerator 12 df in denominator I3 N15 PF gt 30185 PFgt3087B If e ZEIEI7 Nancy Pfenning Eiementary Statistics Luuking attne Big Picture LED 45 Elementary Statistics Looking at the Big Picture Example Testing Relationship 0r Parameters ii Background Research question For all students at a university are Math SATs related to What year they re in El Question How can the question be reformulated in terms of relevant parameters means instead of in terms of Whether or not the variables are related e ZEIEI7 Nancy Pfenning Eiementary Statistics Luuking attne Big Picture LED 46 C 2007 Nancy Pfenning Example Testing Relationship or Parameters El Background Research question For all students at a university are Math SATs related to What year they re in El Response 2 mm mm mm Eiemenhw 5mm mm We aw 7mm Example Testing Relationship or Parameters El Background Research question Do mean eamings differ significantly for three racialethnic groups El Question How can the question he reformulated in terms of relevant variables instead of in terms of Whether or not the means are equal 0mm mm mm Emmy mm mm m at W m 49 Example Testing Relationship or Parameters El Background Research question Do mean eamings differ significantly for three racialethnic groups El Response 2mm mnwmm amnuwsmsm mm We swim mum Lecture Summary Inference for Cat amp Quan ANOVA El Severalsample vs 2sample design I Notation I Compare and contrast t and F statistics I Whatrnakes t or Flarge Variation among means or Within groupsF as ratio of variations How large is large F I Fdegrees offreedorn I F distribution El Role of sample size El El 2mm mm mm ammw 3mm mm um aw We L19 52 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Lecture 18 Continuous Random Variables Tails of the Normal Curve uPreview Two Forms of Inference 16895997 Rule Rule for Tails 90959899 uStandard Normal TailProbability Problems uNonstandard TailProbability Problems 2 mm mm mm swim sums mm mm aw mile Looking Back Review El 4 Stages of Statistics I Data Production discussed in Lectures 14 I Displaying and Summarizing Lectures 512 I Probability El Finding Probabilities discussedin Lectures 1314 El Random Variables introduced in Lecture 15 l Binomial discussed in Lecture 16 I Normal El Sampling Distributions I Statistical Lnference 2mm mm mm gummy smells mm mm aw mm ME 2 Tails of Normal Curve in Inference I Goal Perform inference in 2 forms about unknown population proportion or mean 1 Produce interval that has high probability such as 90 95 or 99 of containing unknown population parameter E Test if proposed value of population proportion or mean is implausible low probabilityl or 5of sample data I Strategy Focus on tails of normal curve in the Vicinity of Z2 or Z2 2mm Nsnwmm Elemenhwstahshcs mm mm swim UEJ 68 95 997 Rule for Z Review For standard normal Z the probability is El 68 that Z takes a value in interval 1 1 El 95 that Z takes a value in interval 2 2 El 997 that Z takes a value in interval 3 3 Need to finetune information for probability at or near 95 2mm mm mm gummy smells mm mm aw mm US 4 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning 90959899 Rule for Standard Normal Z 90959899 Rule Inside Probabilities For Standard normal 2 the is lLooking Ahead This will be useful for con dence intervals El 090 that Z takes a value in interval 1645 1645 El 095 that Z takes a value in interval 1960 1960 El 098 that Z takes a value in interval 2326 2326 El 099 that Z takes a value in interval 2576 2576 90 Looking Back The 689599 7Rule rounded 09544 95 for 2 sds to 0 95 For exactly 95 need 196 sds k Beg rk l ilg b 45 0 16 sol 2 2328 42326 2576 2 576 02007 Naney Pfenning Elementary Statistles Luuklng attne Big Pleture US 5 02007 Nancy Pfenning Elementary statistles Luuklng attne Big Pleture HE E 90959899 Rule Outside Probabilities 90959899 Rule Outside Probabilities For standard normal 2 the is lLooking Ahead This will be useful for hypothesis tests quot I El 005 that Zlt l645 and 005 thatZgt 1645 El 0025 that Z lt l96 and 0025 that Z gt 196 area05 area05 u 001 that Z lt 2326 and 001 that Z gt 2326 area025 area025 i 0005 that Z lt 2576 and 0005 that Z gt 2576 816101 yeah01 Looking Back These follow from the inside probabilities using area fs g005 the fact that the normal curve is symmetric with total area I l i l 1 545 5 l1e45 z 1 960 1960 2 326 2326 2 576 2576 C 2007 Nancy Pfenning Elementary Statistles Luuklng attne Eilg Picture Li a 7 C 2007 Nancy Pfenning Elementary Statistles Luuklng attne Eilg F39lcture Li a a Elementary Statistics Looking at the Big Picture 2 Example Finding Tail Probabilities I Background Refer to sketch area05 area05 area025 area025 area01 area01 Iel 005 area i 71645 0 9545I 71960 1950 C 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 a C 2007 Nancy Pfenning E Example Finding Tail Probabilities I Background Refer to sketch area05 area05 Neat025 area025 area01 aream i area005 o i 1 45 5 H 645l z 1 960 1960 2 325 2 326 72576 2576 II Response PZgt2 326 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 11 Example Finding Tail Probabilities I Background Refer to sketch area05 area05 area025 area025 area01 area01 area005 ar a005 o h P i 71 45 6 L1645 z 1 960 1 960 2826 2 326 72576 I Question What 1s PZlt 1 96 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 12 Elementary Statistics Looking at the Big Picture Example Finding Tail Probabilities I Background Refer to sketch area05 area05 area025 area025 area01 area01 i area005 ar a005 L Vi 45 3 1645 Z 1 960 1 960 72326 2 326 72576 2576 El Response PZlt196 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 14 Example Finding Tail Probabilities I Background Refer to sketch area05 area05 area025 area025 area01 area01 Iel 005 area i 71645 0 9545I 1 960 1950 72576 2576 II Question What is PZgt196 C 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 15 C 2007 Nancy Pfenning E Example Finding Tail Probabilities I Background Refer to sketch area05 area05 Neat025 area025 area01 aream i area005 o i 1 45 5 Ll645 z 1 960 1960 2 328 2 326 72575 2576 II Response PZgt196 by Or Rule 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 17 Example Given Probability Find 2 I Background Refer to sketch area05 area05 area025 area025 area01 area01 area005 ar a005 o h P i 71 45 6 L1645 2 71960 1960 72826 2 326 2575 2576 II Questlon 0 05PZlt 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 1E Elementary Statistics Looking at the Big Picture Example Given Probability Find 2 I Background Refer to sketch area05 area05 area025 area025 area01 area01 i area005 ar a005 L 4amp45 3 1645 Z 1 960 1 960 72328 2 326 72576 2576 El Response 005PZlt 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 2n Example Given Probability Find 2 I Background Refer to sketch area05 area705 area025 area025 area01 area01 Iei 005 area J S 0 i i 71b45 0 L1645 z 71960 1950 72326 2 325 72576 2576 II Questlon 0 005PZgt 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 21 C 2007 Nancy Pfenning E Example Given Probability Find 2 I Background Refer to sketch area05 area05 area025 area025 area01 area01 area005 ar a005 o E i i 713345 6 1e45 z 71 960 1 960 72328 2 326 72576 2576 El Response 0005PZgt 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 23 NonStandard Normal Problems Review To find probability given nonstandard normal x first standardize z 5 then find probability area under 2 curve C 2mm Nancy Pfenning Eiementary Statistics Luuking althe Big Picture L1 8 24 Elementary Statistics Looking at the Big Picture Example Given x Find Probability I Background Women s waist circumferenceX in normal it 32 a 5 area05 area05 area025 area025 area01 area01 i area005 ar a005 0 a g i 1 45 2 J 541 i z 1 960 1860 72 325 2326 2576 2576 El Questlon What 1s PXgt43 0 2mm Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture L1 8 25 3 i J Example Given x Find Probability I Background Women s waist circumferenceX in normal M 32 a 5 area05 area05 area025 area025 area0 i areazm lagg005 area005 I i I 1 45 0 Li 645 z 1 960 1 960 7 336 2 328 2576 2 576 1 Response 0 2mm Nancy Pfenning Eiementaiy Statistics Looking althe Big Picture L1 8 27 C 2007 Nancy Pfenning E Example Given x Find Probability I Background Women s waist circumferenceX in normal it 32 a 5 area05 area05 area025 area025 areaOi area 01 1 area005 ai 005 i I 1 545 0 Li 645i 2 1 1 960 1 960 25 2 325 2576 2 576 El Question What is PXlt23 0 2mm Nancy Pfenning Eiementaiy Statistics Luuking althe Big Picture L1 8 28 Example Given x Find Probability I Background Women s waist circumferenceX in normal M 32 a 5 ai ea05 area05 area025 area025 areaOi areazm i area005 ar a005 0 I i i i i 71 45 0 1645i z 71 1960 1360 392 326 2 32 2 576 2 576 1 Response 0 2mm Nancy Pfenning Eiementaiy Statistics Looking althe Big Picture L1 8 an Elementary Statistics Looking at the Big Picture Example Given x Find Probability I Background Women s waist circumferenceX in normal it 32 a 5 area05 area05 area025 area025 areaz i area01 gt area005 1 ar a005 0 V i i i 71 45 6 31 645i 2 71960 14960 392 395 2 32 2576 2576 El Question What is PXgt39 C ZEIEI7 Nancy Pfenning Eiementaiy Statistics Luuking althe Big Picture L1 8 31 iJ Example Given x Find Probability C 2007 Nancy Pfenning E I Background Women s waist circumferenceX in normal it 32 a 5 area05 i NonStandard Normal Problems Review To find nonstandard x given probability find 2 then unstandardize a H ZO area05 area025 area025 area Oi area01 areai mi g005 l i I a will 1 El Response 1576 2576 ZEIEI7 Nancy Pfenning Eiementaiy Statistics Looking atthe Big Picture HESS C 2mm Nancy Pfenning Eiementaiy Statistics Luuking althe Big Picture Example Given Probability Find x I Background Math SAT score X for population of college students nonnal u 610 o 72 US 34 Example Given Probability Find x I Background Math SAT score X for population of college students nonnal M 610 U e mi Nancy Pfenning Eiementaiy Statistics Luuking althe Big Picture El Response Liaas C ZEIEI7 Nancy Pfenning Elementary Statistics Looking at the Big Picture Eiementaiy Statistics Luuking althe Big Picture LiES7 i J Example Given Probability Find x I Background Math SAT score X for population of college students normal it 610 o 72 area05 area05 area025 area025 areaDi area01 area005 I 005 I i i ri 45 U A645 2 71860 H960 72326 232 0 2mm Nancy Ptenning Eiementaiy Statistics Looking atthe Big Picture Li a 38 C 2007 Nancy Pfenning E Example Given Probability Find x I Background Math SAT score X for population of college students nonnal M 39 a 72 area05 area05 area025 I area025 area Oi areazoi area005 lt i i i 45 b 3i 645i 2 i 960 1 960 2 325 P2 326 o 2575 2576 El Response 0 2mm Nancy Ptennirig Eiernentaiy Statistics Luuking atthe Big Picture Li a 4n Example Given Probability Find x I Background Math SAT score X for population of college students normal it 610 or 72 area05 ai39ea05 area025 area025 ai39ea0i area01 mam 7005 r3 2 575 I 2576 II Question Top half a percent above What score e mi Nancy Ptenning Eiementaiy Statistics Looking atthe Big Picture Li a 4i Elementary Statistics Looking at the Big Picture Example Given Probability Find x I Background Math SAT score X for population of college students nonnal M 610i 0 72 area05 area05 area025 I area025 area Oi ai39eaOi area005 i L 39 H i ei 545 A 645 z 71960 0 1 960 2 326 t2 326 2575 2 576 El Response 0 ZEIEI7 Nancy Ptennirig Eiernentaiy Statistics Looking atthe Big Picture Li a 43 C 2007 Nancy Pfenning E Example Given Probability Find x Example Given Probability Find x El Background Math SAT score X for population of El Background Math SAT score X for population of college students nonnal p 610 a 72 college students nonnal M 610 U 72 area05 area05 lt ai39ea05 area05 meet025 39 meet025 Eek025 area025 mead areazm ai39eaDl arej i 93230 1005 area i 3005 0 i i 1 54s 6 Li 545i I z Ami 45 a A eisgijeo J z 2 326960 32226 72326 Y 39 93912 2575 2 575 2575 12 576 El Response El Question Is PXlt480 more or less than 001 c2uu7 Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture HEM c2uu7 Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture Lima Example Comparing to Given Probability Example Comparing to Given Probability II Background001PZlt2326PZgt2326 Background001PZlt2326PZgt2326 I Question Are the fOHOWlng 0r I Response PZgt24 PZgt24II001 since 24 39 PZgt19 PZgt19001 since 19 I PZlt37 I PZlt37001 s1nce 37 39 PZlt390394 P Zlt 0 4 lo 01 39 0 4 LookingAhead When we perform inference in I 39 39 511106 f 39 f Part 4 some key decisions will be based on how AS Z gets more Heme tall pmbablhty gets a normal probability compares to a set value like 001 or 005 c2uu7 Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture HEM c2uu7 Nancy Pfenning Eiementaiy Statistics Luuking atthe Big Picture LiE4B Elementary Statistics Looking at the Big Picture 9 C 2007 Nancy Pfenning i J Example More Normal Tail Problems El Background Male chest sizes X normal it 3735 or 264 area 05 gt area025 area05 area025 areaOi 353 01 i area005 lt i i i 71645 1 11 545 z i 1 960 1 960 2 32E F 326 2 576 2576 II Question PXgt45 is in what range c 2mm Nancy Pfenning Eiementary Statistics Luuking althe Big Picture Li a 5n Example More Normal Tail Problems El Background Male chest sizes X normal a 3735 a 264 area05 area 05 area025 area025 areaDi area 01 j area J S ar 3005 ii i i 1 545 3 Ni 545i 2 I 960 1 960 r 825 2 326 2 576 2 576 El Response c 2mm Nancy Pfenning Eiementary Statistics Luuking althe Big Picture Li a 52 Example More Normal Tail Problems I Background Female chest sizesX normal a 3515 a 264 area05 area 05 area025 area025 areazoi 7i area 005 i i VI b4 E 1 545 z 1 sea 1 950 gt232C 2 575 2576 El Questlon PXlt288 1s 1n what range c 2mm Nancy Pfenning Eiementary Statistics Luuking althe Big Picture Li a 53 Example More Normal Tail Problems I Background Female chest sizesX normal it 3515 a 264 area05 area 05 area025 areazozs areaz i area 01 7i area005 arga005 i i i 1 i i 43545 0 Mimi z 1 960 1 950 2 325 326 72 576 02 576 El Response e ZEIEI7 Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture Li a 55 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning E Example More Normal Tail Problems Example More Normal Tail Problems I Background Male ear lengtth inches normal I Background Male ear lengtth inches normal M245o017 2245 0217 area05 arEa 05 area05 area 05 area025 area025 area025 area025 areaOi area at area0i area 01 j area at 6005 area005 ar 3005 i 0 lt i4 i i i I 45 1 t b45i z i i 7 545 i ii 545i 2 i 1 960 ii 950 i 4 960 ii 950 2 32s gt2 325 2 325 392 326 72 575 2 575 72 57s 2576 39 0 El Questlon Top 5 A are greater than What value El Response CZEIEI7 Nancy Pfenning Eierneritary Statistics Luukirig atthe Big Picture LiE 5B e mi Nancy Pfenning Eierneritary Statistics Luukirig arthe Big Picture Li a 58 a Example More Normal Tail Problems Example More Normal Tail Problems I Background Female ear lengtth inches nonnal I Background Female ear lengtth inches nonnal a 206 o 017 a 2 2060 017 area05 area 05 area05 area 05 area025 area025 area025 areazozs area0i 365 m areaOi area m 7 397 gt area005 at 3005 area005 ar a005 a 4 0 ii 5 i i i i 45 E 1 545 z i i 7 3545 3 i39iezisi z 1 960 1 950 1 960 1360 2 326 2 325 2325 2 575 2576 52 576 i2 576 39 0 El Questlon Bottom 25 A are less than What value El Response e mi Nancy Pfenning Eierneritary Statistics Luukirig arthe Big Picture ME 59 e ZEIEI7 Nancy Pfenning Eierneritary Statistics Luukirig arthe Big Picture Li 8 Bi Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Exampleszetching Curve with 90959899 Rule Exampleszetching Curve with 90959899 Rule El Background IQsX are normal n 100 a 15 El Background IQsX are normal M 100 0 15 i l l l 39l AS c 1645 Z Z 4860 H960 72326 L2326 2575 2576 El Question How does curve appear El Response 39 e mi Nancy Pfenmng Elementary srausucs Luukmg atthe mg Pmmre m a B2 e mi Nancy Pfenmng Elementary srausucs Luukmg atthe mg Pmmre m a B4 Lecture Summary Tails of Normal Curve II Two forms of inference I Interval estimate I Test if value is plausible II 6895997 Rule and Rule for tails of normal curve I Reviewing normal probability problems I Given x nd probability I Given probability nd x I Focusing on tails of normal curve I Standard normal problems I Nonstandard normal problems e mi Nancy Pfenmng Elementary srausucs Luukmg atthe mg Pmmre L14 as Elementary Statistics Looking at the Big Picture 12 Lecture 24 Confidence Intervals C 2007 Nancy Pfenning Inference for Quantitative Variable lllnference for Means vs Proportions uConstructing CI for Mean SD Known uChecking Normality llDetais of Confidence Interval for Mean uPopulation Standard Deviation Known or Unknown 2 mm Nanm mm Elemenhrv slams Lawmv tithe Blv mam Looking Back Review El 4 Stages of Statistics I Data Production discussed in Lectures 14 I Displaying and Summarizing Lectures 512 I Probability discussed in Lectures 1320 I Statistical Inference 1 categorical discussed in Lectures 2123 D 1 quantitative on dence intervals hypothesis tests a categorical and quantitative n z categorinal u 2 quantitative camamnamma Edmundth mtmanmannmn m2 Inference for Proportions or Means Similarities Inference for Proportions 0r Means Similarities I 3 forms of inference point estimate CI test I Point est unbiased estimator for parameter if 7 I Con dence Interval estimate i margin of error El Sample stat must be unbiased El Pop at least 10 so sd is correct 2 mm Nanm mm Elemenhrv slams Lawmv tithe Blv mam 1 sample statl i lmultiplier Hsd of sample statl El Sample must be large enough so multiplier is correct Note higher confidencea largermultiplier9 wider interval Note larger sample9smaller s d 9naxrower interval Correct interpretation of interval interval related to test Elementary Statistics Looking at the Big Picture I Hypothesis Test Does pmameterroposed value El 3 forms of alternative greater less not equal El 4steps follow 4 processes ofstatis 39c t Dataproduction sampeun iased a large poplen 2 Find sample statistic and standardize is it large 3 Find Pvalueprob ofsample stat this extreme is it small l Draw conclusions reject null hypothesis ifPvalue is small El Pvalue for 2sided alternative twice that for lsided El Cutoff level ct o en 005 is prob onype I Error false pos El Rejection if sample stat far from proposed parameter n arge or sprea all El Type II Error false neg also possible especially for small 2mm Nanm mm Elementalv Statsth lnaklnv the Blv man L244 C 2007 Nancy Pfenning Inference for Proportions or Means Differences I Different summaries for quantitative variables Population mean a Sample mean 37 Population standard deviation 039 EIEIEIEI Sample standard deviation s For proportions sd could be calculated from n andp Standardized statistic not always 2 No easy Rule of Thumb for What n is large enough to ensure normality must examine shape of sample data 2 mm mm mm swim shims mm We aw 7mm Three Types of Inference Problem Mean yearly earnings for sample of 446 students at a particular university was 3776 What is our best guess for the mean eamings of all students at that university Point Estimate What interval should contam mean eamlngs tor all the students Con dence Interval Is this convincing evidence that mean eamings for all the students is less than 5000 Hypothesis Test Eiementaivstatstics makmva heaiv mm i F Fquot mmmmnm mm Behavior of Sample Mean Review Fosample of size n from population Wit mean H sample meanX has I mean H 9 X is unbiased estimator of M sample must be random 2 mm mm mm swim shims mm We aw 7mm Elementary Statistics Looking at the Big Picture Example Checking ifEstimator is Unbiased El Background Anonymous online survey of intro stat students various ages majors at a university produced sample mean eamings El Questions I Is the sample representative of all students at that university Does it represent all college students I Were the values of the variable eamings recorded Without bias mmmmnm mm Eiementaivstatstics makmva heaiv mm C 2007 Nancy Pfenning Example Checking if Estimator is Unbiased Example Point Estimate for M El Background Anonymous online survey of intro stat students various ages majors at a university produced sample mean earnings Responses I Various ages majors 9 El Socioeconomic conditions depend on school 9 I Anonymous online survey9 2mm mnwmm amnuwsmgm mm tithe swim mm El Background In a representative sample of students at a university mean earnings were 3776 El Question What is our best guess for mean earnings of all students at that university mumm mm ammuwsmm makmva heaiv mm 12411 Example Point Estimate for It Three Types of Inference Problem El Background In a representative sample of students at a university mean earnings were 3776 El Response 3 unbiased estimator for H 9 Looking Ahead39 For point estimate we don t need to know s d For con dence intervals and hypothesis tests to quanti how good our point estimate is we must know sigma or estimate it with s This makes an important dijfkrence in procedure 2mm mnwmm amnuwsmgm mm tithe swim 12413 Mean yearly earnings for sample of 446 students at a particular university was I What is our best guess for the mean earnings of all students at that university Point Estimate What interval should contain mean earnings for all the students Con dence Interval Is thls convmcmg ev1dence that mean eammgs for all the students is less than 5000 Hypothesis Test mumm mm ammuwsmm makmva hea v mm 12414 N Fquot Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Inference About Mean Based on 2 or i I a known standardized i is z I a unknown standardized E is I may use 2 if 0 unknown but n large 04 7 z l l r l l l l r l l 74 ea 72 71 0 l 2 3 z or t standardized difference between sample mean and proposed population mean e 2mm Nancy Pfennan Elementary etatlstles Luuklng attne Big F39lcture 03 Inference with t discussed after inference with 2 L24 l5 Confidence Interval for Population Mean with z Use probability results about distribution of sample mean e 2mm Nancy Pfennan Elementary etatlstles Luuklng attne Big F39lcture L24 la Behavior of Sample Mean Review mean X has I mean u a I standard deviation enough n e 2mm Nancy Pfennan Elementary etatlstles Luuklng attne Big F39lcture I shape approximately normal for large For random sample of size n from population with mean M standard deviation 039 sample 9Probability is 095 that X is within 2 ofM L24 l7 Con dence Interval for Population Mean 95 confidence interval for 1 is e I Sample must be unbiased I Population size must be at least 1011 I n must be large enough to justify multiplier 2 from normal distribution e 2mm Nancy Pfennan Elementary etatlstles Luuklng attne Big F39lcture L24 lE Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Guidelines for Sample Mean Approx Normal E Guidelines for Sample Mean Approx Normal Besides examining display consider what shape is expected for the variable s values Can assume shape of X for random samples of size n is approximately normal if Graph of sample data appears normal or Graph of sample data fairly symmetric and n at least 15 or I Graph of sample data moderately skewed and n at least 30 or I Graph of sample data very skewed and n much larger than 30 C 2mm Nancy F39fErlrllrlg Elementary Statistics Luuklng attne Eilg F39lcture L24 la C 2mm Nancy F39fErlrllrlg Elementary Statistics Luuklng attne Eilg F39lcture L24 2n i l 39 7 Example Revisiting Original Question Example Revisiting Original Question I Background Mean yearly earnings fo students at a particular university was Assume population standard deviation 6500 I Background Mean yearly earnings for 446 students at a particular university was 3776 Assume population standard deviation 6500 0 L I Question Assuming sample is representative D Response 95 CI for N Is 30 i 2 what interval should contain population mean earnings Note 446 is large enough to offset rt skewness C 2mm Nancy F39fErlrllrlg Elementary Statistics Luuklng attne Eilg F39lcture L24 2i C 2mm Nancy F39fErlrllrlg Elementary Statistics Luuklng attne Eilg F39lcture L24 23 Elementary Statistics Looking at the Big Picture 5 C 2007 Nancy Pfenning Example C as Range ofPlausible Values El Background Mean yearly earnings for 446 students at a particular university was 3776 Assume population standard deviation 6500 95 con dence interval forll is 3160 4392 El Question Is 5000 a plausible value for population mean earnings 2mm mnwmm amnuwsmgm mm mm swim 1242 Example C as Range ofPlausible Values El Background Mean yearly eamings for 446 students at a particular university was 3 776 Assume population standard deviation 6500 95 con dence interval for is 3160 4392 El Response Looking Ahead This kind of decision is addressed more formally and precisely with a hypothesis test 2mm mm mm ammw Statstics mm um aw mm mm Example Role of Sample Size in CI El Background Mean yearly earnings for 446 students at a particular university was 3776 Assume population standard deviation 6500 95 con dence interval for it is 3776 i 616 El Question What would happen to the CI if n were one fourth the size 111 instead of 446 2mm mnwmm amnuwsmgm mm mm swim 12427 Example Role of Sample Size in CI Elementary Statistics Looking at the Big Picture El Background Mean yearly eamings for 446 students at a particular university was 3 776 Assume population standard deviation 6500 95 con dence interval for is 3776 i 616 El ResponseDivide n by 49 2mm mm mm ammw Statstics mm um aw mm 12429 C 2007 Nancy Pfenning E Example Other Levels of Con dence Example Other Levels of Con dence I Background Mean yearly earnings for 446 I Background Mean yearly earnings for 446 students at a particular university was 3776 students at a particular university was 3776 Assume population standard deviation 6500 Assume population standard deviation 6500 A 95 confidence interval for M A 95 confidence interval for M 6500 6500 3776 2m 3776 I 616 31604392 3776 2m 3776 i 616 31604392 II Question How would we construct intervals I Response at 90 or 99 confidence 7 Other Levels of Con dence Other Levels of Con dence Review Inside probs correspond to various multipliers Con dence level 95 uses multiplier 2 Other levels use other multipliers based on normal curve More precise multiplier for 95 is 196 instead of 2 Level M u Iti plier 90 1645 399 gt 95 1960 95 0 k 33 98 2326 l rx 0 i 1 45 10 1645l z 99 A 2576 4960 1960 2326 2826 72576 2576 e 2mm Nancy Pfenning Eiementary Statistics Luuking althe Big Picture L24 33 e mi Nancy Pfenning Eiementary Statistics Luuking althe Big Picture L24 34 Elementary Statistics Looking at the Big Picture 7 C 2007 Nancy Pfenning Example Other Levels of Con dence El Background Mean yearly earnings for 446 students at a particular university was 3776 Assume population standard deviation 6500 El Question What are 90 and 99 con dence intervals for population mean earnings 2mm mmmm ElemenDHStaushcs mm tithe swim l2435 Example Other Levels of Con dence El Background Mean yearly eamings for 446 students at a particular university was 3 776 Assume population standard deviation 6500 El Response Interval is 3776 i multiplier 6 l 90 n CI V 446 I 99quotn CI Tradeo higher level of c0n dence9less precise 2mm mm mm ammw slums mm um aw mm L2437 Wider Intervals 69 More Con dence Consider illustration of many 90 con dence intervals in the long run 18 in 20 should contain population parameter If they were widened to 95 intervals multiply sd by 2 instead of 1645 then they d have a higher probability 19 in 20 of capturing population parameter 2mm mmmm ElemenDHStaushcs mm tithe swim um Wider Intervals 6 More Con dence Elementary Statistics Looking at the Big Picture Whim 5 H i mmmmnm mm C 2007 Nancy Pfenning Interpretation of XX Con dence Interval We are XX con dent that the interval contains the unknown parameter XX intervals longrun probability of capturing the unknown parameter is XX 2mm mnwmm amnuwsmsm mm tithe swim 12442 Example Interpreting Con dence Interval El Background A 95 con dence interval for mean US household size L is 2166 2714 El Question Which are true T and which false F 7 Probability is 95 that L is in the interval 2166 2714 95 ofhousehold sizes are in the interval 2166 2714 Probability is 95 that E is in the interval 2166 2714 We re 95 con dent that f is in interval 2166 2714 We re 95 con dent that U is in interval 2166 2714 The probability is 95 that We produced an interval which contains ll 2mm mm mm ammw Sialstics mm um aw mm 12443 Example Interpreting Con dence Interval El Background A 95 confidence interval for mean US household size 1 is 2166 2714 El Response Probability is 95 that U is in the interval 2166 2714 95 ofhousehold sizes are in the interval 2166 2714 Probability is 95 that c is in the interval 2166 2714 We re 95 con dent that if is in interval 2166 2714 We re 95 con dent that IL is in interval 2166 2714 The probability is 95 that We produce an interval which contains Ha 2mm mnwmm amnuwsmsm mm tithe swim 12446 Elementary Statistics Looking at the Big Picture Lecture Summary Inference for Means Con dence Interval El Inference for means vs proportions I Similarities many I Differences population sd may be unknown Constructing CI for mean with z pop sd known Checking assumption of normality Role of sample size Other levels of confidence Interpreting the confidence interval EIEIEIEIEI 2mm mm mm ammw Sialstics mm um aw mm L19 4a C 2007 Nancy Pfenning Lecture 4 Designing Studies Focus on Sample Surveys ulssues for any Study Design ulssues in Design of Sample Survey Questions 2 mm mm mm Elementary shims mm We aw 7mm Looking Back Review El 4 Stages of Statistics I Displaying and Summarizing I Probability I Statistical Inference 2mm mm mm Eiementaiy Statstics mm mm aw mm in Looking Back Review El Types of Study Design varia C I Experiment researchers control explanatory I Observational study values occur naturally El Special case sample surveys o en selfreported l El Two steps in Data Production I Obtain an unbiased sample summary 0 sa I Assess variables values to obtain unbiased El Design survey questions to assess Values Without bias 2 mm mm mm Elementary shims mm We aw 7mm in Example Formulating a Survey Question El Background A popular 2005 movie sparked speculation how common is it for a 40year old male to be a virgin El Question Assuming you had a representative sample of 40yearold males what survey question would you ask to nd out what proportion are virgins Students can jot down question amp discuss a er covering issues in survey question design 2mm mm mm Eiementaiy Statstics mm mm aw mm m Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Sample Survey Design Issues to Consider Example Open vs closed questions El Open vs closed questions El Unbalanced response options El Leading questions or planting ideas with questions El Complicated questions El Sensitive questions El Hardtode ne concepts 2mm mnwmm Eiemenhwstahshcs mm tithe gimme L45 El Questions 1 What kind of question is this a open b closed 2 What is an open question 2mm mm mm ammw Statstics mm um aw mm L45 Example Open vs closed questions De nitions El Responses 1 What kind of question is this a open b closed 2 What is an open question 2mm mnwmm Eiemenhwstahshcs mm tithe gimme ma El An open question does not have a xed set of response options El A closed question either provides or implies a xed set of possible responses 2mm mm mm ammw Statstics mm um aw mm L4H Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Overly restrictive options Example Overly restrictive options El Background A neuroscientist asked survey respondents How o en do you dream in color Answer alwayssometimesnever El Question What is the most important improvement that should be made to this survey question 2mm mnwmm amnuwsmms mm tithe swim mu El Background A neuroscientist asked survey respondents How often do you dream in color Answer alwayssometimesnever El Response mmu mm mm ammwsmm makinvanheatv We L412 Example Unbalanced Response Options Example Unbalanced Response Options El Background 91 of Americans surveyed rated their own health as good to excellent El Questions I Is this result surprising to you I If so does it seem unexpectedly high or low 2mm mnwmm amnuwsmms mm tithe swim L413 El Background 91 of Americans surveyed rated their own health as good to excellent El Response mmu mm mm ammwsmm tnakmvanheaw We L415 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Unbalanced Response Options El Background 91 of Americans surveyed rated their own health as good to excellent Options provided were Excellent Very Good Good Fair Poor El Question Now is the result surprising 2mm mnwmm Eiemenuwsuusucs mm mm swim L416 Example Unbalanced Response Options El Background 91 of Americans surveyed rated their own health as good to excellent Options provided were Excellent Very Good Good Fair Poor El Response mmmmnm mm ammwsmm makmvanheaiv mm ma Example Deliberate bias El Background The following question was posted on wwwahumanrightcom Ifmy child or my spouse were assaulted I would choose one Run away and hope my kid or spouse can keep up 7 Be a good Witness so I can tell the cops What happened later Try to convince the attacker to stop through verbal PBISHZSIOH 4 Fight to stop the attack El Question Do we know what response the surveyer wants us to choose 4 2mm mnwmm Eiemenuwsuusucs mm mm swim L419 Elementary Statistics Looking at the Big Picture Example Deliberate bias El Background The following question was posted on wwwahumanrightcom If my child or my spouse were assaulted I would Run away and hope my kid or spouse can keep up Be a good Witness so I can tell the cops What happened later Try to convince the attacker to stop through verbal PBISHZSIOH 4 Fight to stop the attack El Response We are obviously supposed to c oosei H mmmmnm mm ammwsmm damning mm L421 C 2007 Nancy Pfenning Deliberate Bias If it s clear what response the surveyer wants then the results are not useful from a statistical standpoint 2mm mnwmm amnuwsmsm mm mm gimme L422 Example Complicated question El Background A telephone surveyer asked a homemaker to agree or disagree with this I don t go out of my way to purchase lowfat foods unless they re also low in calories El Question How can this survey question be improved 2mm mm mm ammw Stalstics mm um aw mm L423 Example Complicated question El Background A telephone surveyer asked a homemaker to agree or disagree with this I don t go out of my way to purchase lowfat foods unless they re also low in calories El Response 2mm mnwmm amnuwsmsm mm mm gimme L425 Example A controversial question Elementary Statistics Looking at the Big Picture El Background Anonymous FA Youth Survey given to 6m12Lh public school students asked How old were you when you first I got suspended from school I got arrested I carried a handgun etc Choose never have 10 or younger 11 12 17 El Questions I Why did parents object I Why was the question worded this way 2mm mm mm ammw Stalstics mm um aw mm L425 C 2007 Nancy Pfenning Example A controversial question El Background Anonymous FA Youth Survey given to 6 12 h public school students asked How old were you when you rst I got suspended from school I got arrested I carried a handgun etc Choose never have 10 oryounger 11 12 17 El Responses 2mm mnwmm amnuwsmgm mm mm mm L475 Example Keyboards for Sense ofAnonymity El Background A stats computer tutor was piloted in a class where students consented to be identi ed by name Still one student lled in the text boxes with obscenities El Question Why did the student write inappropriately in the computer lab and not on his hardcopy homeworks or exams 2mm mm mm ammw Statstics mm um aw mm mu Example Keyboards for Sense ofAnonymity El Background A stats computer tutor was piloted in a class where students consented to be identi ed by name Still one student lled in the text boxes with obscenities El Response This tendency is used to researchers advantage when seeking responses to sensitive questions 2mm mnwmm amnuwsmgm mm mm mm L432 Example HardtoDe ne Concepts Elementary Statistics Looking at the Big Picture El Background A survey found 19 of Americans believe money can bu appiness I Robert Frost Happiness makes up in height for What it lacks in length I Albert Camus But What is happiness except the simple harmony between a man and the life he leads El Questions I By Frost s de nition can money buy happiness I By Camus s definition can money buy happiness I What de nition of happiness Were respondents using 2mm mm mm ammw Statstics mm um aw mm L433 C 2007 Nancy Pfenning Example HardtoDe ne Concepts Example Formulating a Survey Question El Background A survey found 19 of Americans believe money can buy happiness I Robert Frost Happiness makes up in height for what it lacks in length I Albeit Camus But what is happiness except the simple harmony between a man and the life he leads El Responses I Frost I Camus I Respondenw 2mm mnwmm amnuwsmm mm tithe swim L435 El Background Earlier we asked Assuming you had a representative sample of 40year old males what survey question would you ask to nd out what proportion are virgins El Question Are you satis ed with the phrasing of your question and if not how would you rephrase it 2mm mm mm ammw Stalslics mm um aw We ma Example Formulating a Survey Question Issues to Consider for Any Study Design El Background Earlier we asked Assuming you had a representative sample of 40year old males what survey question would you ask to nd out what proportion are virgins El Response Consider I Open or closed I I f closed what response options are provided I Is question designed to elicit honest responses I Is the concept well de ned 2mm NW amnuwsmm mm tithe swim L437 a El Errors in Study s Conclusions 2mm mm mm ammw Stalslics mm um aw We Lass Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Sample Size and Study Design El Background Researchers want to know if stronger sunscreens cause more time in sun They could design an observational study or an experiment to test this El Question Which is better using 10 students or 100 students 2mm mnwmm Eiemenhwsuusucs mm tithe swim ma Example Sample Size and Study Design El Background Researchers want to know if stronger sunscreens cause more time in sun They could design an observational study or an experiment to test this El Response It depends I If study is flawed obs study with confounding variables or poorly designed experiment9 I If study is well designed9 2mm mm mm ammw Statstics mm um aw mm L441 Issues to Consider for Any Study Design El Sample size El Error nStudy s Conclu ons 2mm mnwmm Eiemenhwsuusucs mm tithe swim L442 Example Two Types of Error Elementary Statistics Looking at the Big Picture El Background A study tested effectiveness of radar guns to identify speeders El Question What are the two possible errors in the study s conclusions and the potential harmful consequences of each Note the study either concludes that the guns work properly or that they do not 2mm mm mm ammw Statstics mm um aw mm L443 C 2007 Nancy Pfenning Example Two Types of Error Example Sample Size and Error El Background A study tested effectiveness of radar guns to identify speeders El Response 2mm mnwmm Eiemenhwstausucs mm tithe swim L416 El Background A study tested effectiveness of radar guns to identify speeders El Question Which error is more likely to be made if only a small sample of guns is tested 2mm mm mm ammw Statstics mm um aw mm we Example Sample Size and Error Example Errors in Home Drug Testing El Background A study tested effectiveness of radar guns to identify speeders El Response 2 mm mm mm Elementary shims mm tithe aw 7mm El Background A study discussed limitations and risks in the use of home drug testing kits El Question What are the two possible errors in a drug test s conclusions and the potential harmful consequences of each 2mm mm mm ammw Statstics mm um aw mm L449 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Errors in Home Drug Testing El Background A study discussed limitations and risks in the use of home drug testing kits El Response 2mm mnwmm amnuwsmsm mm tithe swim L451 Lecture Summary Sample Surveys Elementary Statistics Looking at the Big Picture Open vs closed questions Unbalanced response options Leading questions Complicated questions Sensitive questions Hardtode ne concepts EIEIEIEIEIEIEI Issues for any study design I Sample size I Errors in st39udy s conclusions 2mm mm mm ammw Statstics mm um aw mm L452 Lecture 14 Finding Probabilities More General Rules uGeneral OI Rule uConditional Probability General And Rule uTwo Types of Error ulndependence 2 mm mm mm Elementary slums mm We aw 7mm C 2007 Nancy Pfenning Looking Back Review El 4 Stages of Statistics I Data Production discussed in Lectures 14 I Displaying and Summarizing Lectures 512 I Probability El Random Variables El Sampling Distributions I Statistical Inference 2mm mm mm ammw slums mm um aw mm L142 Basic Probability Rules Review Non Overlapping Or Rule For any two nonoverlapping events A and B PA or BPAPB Independent And Rule For any two independent events A and B PA and BPAxPB 2mm mnwmm ElemenDHStaushcs mm am whenquot L143 More General Probability Rules Elementary Statistics Looking at the Big Picture I Need Or Rule that applies even if events overlap I Need And Rule that applies even if events are dependent I Consult twoway table to consider combinations of events when more than one variable is involved 2mm mm mm ammw slums mm um aw mm L144 C 2007 Nancy Pfenning Example Parts of Table Showing Or and And Example Parts of Table Showing Or and And El Background Professor notes gender female or male and grade A or not A for students in class El Questions What part of a twoway table shows I Students Who are female and get an A I Students Who are female or get an A A not A Total Female 015 0 45 0 60 Male 0 10 0 30 0 40 Total 0 25 0 75 l 00 2mm mnwmm amnuwsmsm mm tithe swim L145 El Background Professor notes gender female or male and grade A or not A for students in class El Responses I Students Who are female and get an A table on if I Students Who are female or get an A table on if not A Total 2mm mm mm ammw Statstics mm um aw mm L147 Example Intuiting General Or Rule El Background Professor reports probability of getting an A is 025 probability of being female is 060 Probability of both is 015 El Question What is the probability of being a female or getting an A 2mm mnwmm amnuwsmsm mm tithe swim ma Example Intuiting General Or Rule El Background Professor reports probability of getting an A is 025 probability of being female is 060 Probability of both is 015 El Response 2mm mm mm ammw Statstics mm um aw mm m m Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Intuiting General Or Rule General Or Rule General Addition Rule El Response Illustration with A notA Total Fem ale Male mi i i not A Total Total 2mm mnwmm amnuwsmsm mm tithe swim mm For any two events A and B PAor PA PB PAandB 0 if no overlap A Closer Look In general the word or in probability entails addition 2mm mm mm ammw Stalstics mm um aw mm L1412 Example Applying General Or Rule Example Applying General Or Rule El Background For 36 countries besides the US who sent troops to Iraq the probability of sending them early by spring 2003 was 042 The probability of keeping them there longer still in fall 2004 was 078 The probability of sending them early anal keeping them longer was 033 Question What was the probability of sending troops early or keeping them longer El 2mm mnwmm amnuwsmsm mm tithe swim L141 El Background For 36 countries besides the Us who sent troops to Iraq the probability of sending them early by spring 2003 was 042 The probability of keeping them there longer still in fall 2004 was 078 The probability of sending them early anal keeping them longer was 033 El Response 2mm mm mm ammw Stalstics mm um aw mm L1415 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Basic Probability Rules Review 39t i Example When Probabilities Can 1 Simply be Multiplied Review NonOverlapping 0r Rule For any two nomoverlapping events A and E El Background In a child 5 pocket are 2 quarters and 2 nickels He randomly picks a coin does not PA 0r BPAPB replace it and picks another Independent And Rule For any two El Response To find the probability of the rst and independent events A and B the second coin being quarters we can t multiply 05 P A and BPAXPB by 05 because after the rst coin has been removed the probability of the second coin being a quarter is not 05 it is 13 if the first coin was a quarter 23 if the rst was a nickel e ZEIEI7 Nancy Pfenning Elementary Statisties taaking attne Big Picture Ll4 W e ZEIEI7 Nancy Pfenning Elementary Statisties Laaking attne Big Picture Ll4 lE L iii Example When Probabilities Can 1 Simply be De nition an d Notation Multiplied Possibilities iOi isi seieciion Conditional Probability of a second event given a first event is the probability of the second event occurring assuming that the I Probability of a qiiarter is 2412 I rSt eVent has occurred PB given A denotes the conditional Possibilities for 2nd selection pTObability 0f eVent B occurring giVen i that event A has occurred l LookingAhead Conditional probabilities help us Probability of a quarter is 13 Probability of a quarter is 23 I handle dependent events it 1st selection was a quarter i it 1st selection was a nickel e ZEIEI7 Nancy Pfenning Elementary Statisties taaking attne Big Picture HA is e ZEIEI7 Nancy Pfenning Elementary Statisties taaking attne Big Picture MA in Elementary Statistics Looking at the Big Picture 4 C 2007 Nancy Pfenning Example Intuttmg the General And Rule I Background In a child s pocket are 2 quarters and 2 nickels He randomly picks a coin does not replace it and picks another I Question What is the probability that the first and the second coin are quarters C 2mm Nancy Ptenning Elementary Statistics Luuking attle Big Picture Ll4 Zl E Example Intuttmg the General And Rule I Background In a child s pocket are 2 quarters and 2 nickels He randomly picks a coin does not replace it and picks another I Response probability of first a quarter 24 times conditional probability that second is a quarter given first was a quarter 13 C 2mm Nancy Ptenning Elementary Statistics Luuking attle Big Picture LN 23 Example Intuttmg the General And Rule the times Wiien isi CUlll is Ouai tei 739 Di 2m Colii is also OtallEl C 2mm Nancy Ptenning mentary Statistics Luuking attle Big Picture Example Intuttt39ng General And Rule with Two Way Table I Background Surveyed students classified by sex and Whether or not they have ears pierced Ears Ears not piercedi pierced Female 270 30 300 Total Male 20 180 200 Toiai 290 210 500 El Question What are the following probabilities I PM being male I PE given M having ears pierced given male I PM and E being male and having ears pierced C 2mm Nancy Ptenning Elementary Statistics Luuking attle Big Picture LN 25 Elementary Statistics Looking at the Big Picture Two Way Table Example Intuiting General And Rule with Ears Ears not pierced pierced Femaie 270 30 300 Total Male 20 180 200 Total 290 210 500 El Response PM I PE given M I PM and E C 2mm Nancy Pfenning Eiementary Statistics Luuking aims Big Picture I Background Surveyed students classi ed by sex and whether or not they have ears pierced Restricted to Male row LMZE C 2007 Nancy Pfenning Example Intuiting General And Rule with Two Way Table El C 2mm Nancy Pfenning Eiemenrary Statistics Looking at Background Surveyed students classi ed by sex and whether or not they have ears pierced Ears Ears not piercedi pierced Female 270 30 300 Total Male 20 180 200 Tom 290 210 500 Response I PM I PE given M I or PM and E Note PliIana39E 7E PMP 20400580272 mm a Big Picture Restricted to Male row Rule For any two events A and B PABPAPB given A I l PB if independent I entails multiplication C 2mm Nancy Pfenning Eiementary Statistics Luuking aims Big Picture General And Rule General Multiplication A Closer Look In general the word and in probability Li43i Example Applying General And Rule El II Question What are the following probabilities C Background Studies suggest lie detector tests are well below perfection 80 of the time concluding someone is a spy when heshe actually is 16 of the time concluding someone is a spy when heshe isn t We ll assume 10 of 10000 govt employees are spies I Probability of being a spy and being detected as one I Probability of not being a spy but detected as one I Overall probability of a positive lie detector test 2mm Nancy Pfenning Eiementary Statistics Luuking aims Big Picture LN 33 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Applying General Anal Rule El Background Studies suggest lie detector tests are well below perfection 80 of the time concluding someone is a spy when heshe actually is 16 of the time concluding someone is a spy when heshe isn t We ll assume 10 of 10000 govt employees are spies Note PD given S0s PD given not S016 PS0001 Pnol S0 999 El Response I being a spy and being detected as one I not being a spy and detected as one I Overall probability Or Rule 2mm mnwmm Eiemenurysuusucs mm 31th WNW was Example Or Probability as Weighted Average of Conditional Probabilities El Background Studies suggest lie detector tests are well below perfection 80 of the time concluding someone is a spy when heshe actually is 16 of the time concluding someone is a spy when heshe isn t We ll assume 10 of 10000 govt employees are spies El Question Should we expect the overall probability of being detected as a spy PCD to be closer to PD given S080 or to PD given not S016 2mm mm mm ammw 3mm mm um aw mm mun Example Or Probability as Weighted Average of Conditional Probabilities El Background Studies suggest lie detector tests are well below perfection 80 of the time concluding someone is a spy when heshe actually is 16 of the time concluding someone is a spy when heshe isn t We ll assume 10 of 10000 govt employees are spies El Response Vast majority are not spies9expect PCD closer to In fact PCD was found to be 016064 2mm mnwmm Eiemenurysuusucs mm 31th WNW mug General And Rule Leads to Rule of Conditional Probability Recall For any two evens A and B PA and BPAxPB given A Rearrange to form Rule of Conditional Probability PB given A PA and B PA 2mm mm mm ammw 3mm mm um aw mm mm Elementary Statistics Looking at the Big Picture Example Applying Rule of Conditional Probability El Background For the lie detector problem we have Probability of being a spy PS0001 Probability of spies being detected PD given S080 Probability of nonspies detected PD given not S0 16 Probability of being a spy and detected PS and D00008 Overall probability ofpositive lie detector PD16064 Question If the liedetector indicates an employee is a spy What is the probability that heshe actually is one El 2mm mnwmm Eiemenurysuusucs mm mm swim mm C 2007 Nancy Pfenning Example Applying Rule of Conditional Probability El Background For the lie detector problem we have I Probability ofbeing a spy PS001 I Probability of spies being detected PD given S080 I Probability of nonspies detected PD given not S416 I Probability ofbeing a spy and detected PD and 410008 I Overall probability of positive lie detector PD0 16064 Response PS given DPD and S PD Note PSgiven D is very difkrentfrom PD given S El 2mm mm mm ammw 3mm mm um aw mm mm Two Types of Error in Lie Detector Test 1st Type of Error Conclude employee is a spy when heshe actually is not 2 d Type of Error Conclude employee is not a spy when heshe actually is 2mm mnwmm Eiemenurysuusucs mm mm swim 1144a Example Two Types of Error in Lie Detector Test Elementary Statistics Looking at the Big Picture El Background For the lie detector problem we have I Probability of spies being detected PD given S080 I Probability of nonspies detected PD given not S 16 El Questions I What is probability of 1st type of error conclude employee is spy when heshe actually is not I What is probability of 2quotquot1 type of error conclude employee is not a spy when heshe actually is 2mm mm mm ammw 3mm mm um aw mm m 47 C 2007 Nancy Pfenning i i Exam le Two T es 0 Error in Lie Detector p yp f Testing for Independence Test The concept of independence is tied in with I Background For the he detector problem we have conditional probabilities I Probability of spies being detected PD given S080 I I Probability of nonspies detected PD given not S0 16 L00kmg Ahead Much ofstansncs concerns Itself with whether or not two events or two variables El Responses are dependent related I 1 type PD given not S I 2nd type Pnot D given S Eiementary Statistics Luuking atthe Big Picture Li4 5n e mi Nancy Pfenning Eiementary Statistics Luuking atthe Big Picture LN 43 e mi Nancy Pfenning Example Intuiting Conditional Probabilities When Events Are Dependent Example Intuiting Conditional Probabilities When Events Are Dependent I Background Students are classified according to gender M or F and ears pierced or not E or not E Responses I Background Students are classified according to gender M or F and ears pierced or not E or not E El El Questions I Should gender and ears pierced be dependent or ind If I 7 Expect 7 7 lower than 777 dependent which should be lower PE or PE given M because fewer males have pierced ears I What are the above probabilities and which is lower I PE given M PE Ears Ears not Total piercedi pierced Female 270 30 300 Ears Ears not pierced pierced Femaie 270 30 300 Total Male 20 180 200 Male 20 180 200 Tota 290 210 500 Eiementary Statistics Luuking atthe Big Picture Tota 290 210 500 e mi Nancy Pfenning L e e LMSi CZEIEI7 Nancy Pfenning Li453 Elementary Statistics Looking at the Big Picture Example I ntuiting Conditional Probabilities When Events Are Independent C 2007 Nancy Pfenning El Background Students are classi ed according to gender M or F and Whether they get an A in Stats Questions I Should gender and getting an A or not be dependent or ind How should PA and PA given F compare I What are the above probabilities and which is higher A not A Total El Female 015 0 45 0 60 Male 0 10 0 30 0 40 Total 0 25 0 75 1 00 2mm mnwmm Eiemenuwsuusucs mm tithe swim L1454 Example I ntuiting Conditional Probabilities When Events Are Independent El Background Students are classi ed according to gender M or F and Whether they get an A in Stats El Responses l Gender and grade should be 7 i should have PA77PA given F l PA 9 015 045 010 030 0 25 0 75 100 2mm mm mm ammw Stalstics mm um aw mm was Independence and Conditional Probability Rule A and B independent9PBPB given A Test PBPB given A9A and B are independent PB PB given A9A and B are dependent Independente regular and conditional probabilities are equal occurrence of A doesn t affect probability of B 2mm mnwmm Eiemenuwsuusucs mm tithe swim L1457 Independence and Product of Probabilities Elementary Statistics Looking at the Big Picture Rule Independent9PA and BPAgtltPB Test PA and B PAxPB9independent PA and B PAxPB9dependent Independent pr0bability of both equals product of individual probabilities 2mm mm mm ammw Stalstics mm um aw mm L145a C 2007 Nancy Pfenning Table of Counts Expected if Independent Example COW Expected lflndependem I For A B independent El Background Students are classified according to PA and BPAgtltPB gender and ears pierced or not A table of expected I This Rule dictates what counts would counts 174 293004 091135 been pmduced39 appear in twoway table ifthe variable 31233Z titif f12 2 l A or not A is independent of the variable W B 0139 not B 300 I If independent count in category 200 combination A and B must equal Tm l 29quot 2 50quot total in A times total in B divided by overall total in table I Question How different are the observed and expected counts e 2mm Nancy Pfenning Elementary Statistics Luuking alme Big Picture L14 an e 2mm Nancy Pfenning Elementary Statistics Luuking alme Big Picture L14 El Example Counts Expected if Independent Example Counts Expected if Independent I Background Students are classified according to I Background Students are classified according to gender and ears pierced or not A table of expected gender and grade A or not A table of expected counts 174 w etc has been produced counts 15 m etc has been produced Counts expected It gender ano Counts actually 1 00 pierced ears were independent observed E E Tom Exp A not A Total Obs A not A Total WM F 15 45 60 F 15 45 60 M M 10 3o 40 M 10 3o 40 We Total 25 75 100 Total 25 75 100 El Response Observed and expected counts are very different 270 VS 174 20 VS 116 etc because I Question How different are the observed and expected counts e 2mm Nancy Pfenning Elementary Statistics Luuking alme Big Picture L14 as e mi Nancy Pfenning Elementary Statistics Luuking alme Big Picture L14 B4 Elementary Statistics Looking at the Big Picture 11 C 2007 Nancy Pfenning Example Counts Expected if Independent Ledlll e Summary Finding Probabilities More General Rules El Background Students are classi ed according to gender and grade A or not A table of expected 5 General 0139 Rule counts 15 etc has been produced El Conditional Probability El General And Rule El Two Types of Error El Independence I Testing for independence I Rule for independent evens I Counts ex ected if inde endent El Response Counts are identical because P P 2mm mnwmm Eiemenuwsuusucs mm tithe swim was mumm mm ammmsmm makmva heaiv mm mm Elementary Statistics Looking at the Big Picture 12 C 2007 Nancy Pfenning Looking Back Review Lecture 21 El 4 Stages of Statistics I Data Production discussed in Lectures 14 Inference for Categorical Varlable I Displaying and Summarizing Lectures 512 Confidence Intervals Probability discussed in Lectures 1320 I Statistical Inference 13 Forms of Inference uProbability vs Con dence 539 1 quantitative El ale nical and 39uanlitatsv uConstructIng Confidence Interval gt 1 r El v categorical uSample Size Level of Confidence E Z quantitative em W Four Processes Of Statistics Summarizing Categorical Sample Data Review P0plllati0n L PRODUCE DATA What proportion o id ntiatgelll reikfast Assume we the day ofthe survey 17 h 446 055 Looking Back In Part 2 we summarized sample data for single variables or relationships Looking Ahead In Part 4 our goal is to go beyond sample data and draw conclusions about the larger population from which the sample was obtained Assume we only know what 39s true for the sample what can we in fer about the larger population 2mm NamHenan Elemenhwstaushcs mm mm serum in a mmu unw minim HangmanStatsth mammal mm L214 Elementary Statistics Looking at the Big Picture 1 C 2007 Nancy Pfenning Three Types of Inference Problem Behavior of Sample Proportion Review In a sample of 446 students 055 ale breakfast 1 What is our best guess for the population proportion of students who eat breakfast Point Estimate 2 What interval should contain the population proportion of students who eat breakfast Con dence Interval Fquot Is the population proportion of students who eat breakfast more than half 50 Hypothesis Test 2mm mnwmm amnuwsmsm mm tithe swim 1215 Fosample of size n from population wrt p 1n category of interest sample proportion 13 has I mean p 9 33 is unbiased estimator of p sample must be random 2mm mm mm gummy Sialstics mm um aw mm L216 Example Checking if Estimator is Unbiased Example Checking ifEstimaior is Unbiased El Background Survey produced sample proportion of intro stat students various ages and times of day at a university who d eaten breakfast El Questions I Is the sample representative of all college students All students at that university I Were the values of the variable breakfast or not recorded Without bias 2mm mnwmm amnuwsmsm mm tithe swim 1217 El Background Survey produced sample proportion of intro stat students of various years in classes meeting at various times of day at a university who d eaten breakfast El Responses I Differences among college cafeterias etc9 I Question not sensitive9 2mm mm mm gummy Sialstics mm um aw mm L219 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Point Estimate for p Example Point Estimate for p El Background In a representative sample of students 055 ate breakfast El Question What is our best guess for the proportion of all students at that university who eat breakfast 2mm mnwmm amnuwsmgm mm tithe swim 121m El Background In a representative sample of students 055 ate breakfast El Response 13 unbiased estimator for p9 7 is best guess forp mmmmnm mm ammwsmms makmvanheaiv mm L21 2 Example Point Estimate Inadequate Example Point Estimate Inadequate El Background Our best guess for population proportion p eating breakfast is sample proportion 055 El Question I Are we pretty sure the population proportion is 055 2mm mnwmm amnuwsmgm mm tithe swim R113 El Background Our best guess for population proportion p eating breakfast is sample proportion 055 El Response mmmmnm mm ammwsmms damning mm L21 15 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Point Estimate Inadequate Example Point Estimate Inadequate El Background Our best guess for population proportion p eating breakfast is sample proportion 055 El Question I By approximately What amount is our guess of 9 2mm mnwmm mmmsmm mm tithe mm R116 El Background Our best guess for population proportion p eating breakfast is sample proportion 055 El Response mumm mm ammmum mmumaw mm L21 1a Example Point Estimate Inadequate Example Point Estimate Inadequate El Background Our best guess for population proportion p eating breakfast is sample proportion 055 El Question I Are we pretty sure population proportion is more than 050 2mm mnwmm mmmsmm mm tithe mm R1 19 El Background Our best guess for population proportion p eating breakfast is sample proportion 055 El Response mumm mm ammmum mmumaw mm L2121 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Three Types of Inference Problem E Beyond a Point Estimate In a sample of 446 students 055 ate breakfast 1 What is our best guess for the population proportion of students who eat breakfast Sample proportion from unbiased sample is best estimate for population proportion Looking Ahead For point estimate we don t POint EStimate need sample size or info about spread These 2 What interval should contain the population are requiredfo and proportion of students who eat breakfast hypothesis testsWd our point estimate is Confidence Interval 3 Do more than half 50 of the population of students eat breakfast Hypothesis Test C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg F39lCturE t2l 22 C 2mm Nancy F39fErlrllrlg Elementary Statlstles Lunklng attne Eilg F39lCturE t2l 23 2 Probability vs Con dence Example Probability Statement I Background If students pick numbers from 1 to 20 at random p005 should pick 7 I Con dence given sample proportion For quot2400 33 has what is a range of plausible values for I mean 005 population proportion I sd W 001 I shape approx normal I Question What does the 95 part of the 6895997 Rule tell us about 15 I Probability given population proportion how does sample proportion behave C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg F39lCturE t2l 25 C 2mm Nancy F39fErlrllrlg Elementary Statlstles Luuklng attne Eilg F39lCturE t2l 2e Elementary Statistics Looking at the Big Picture 5 Example Probability Statement I Background If students pick numbers from 1 to 20 at random p005 should pick 7 For n400 13 has I mean 005 sd W 001 l shape approx normal I Response C 2mm Nancy Pfenning Eiementary Statistics Looking atthe Big Picture L21 28 C 2007 Nancy Pfenning E Example Probability Statement Looking Ahead This statement about sample proportion is correct but not very useful for practical purposes In most reallife problems we want to draw conclusions about an unknown population proportion C 2mm Nancy Pfenning Eiementary Statistics Looking althe Big Picture L21 2a Example How Far is One from the Other I Background Suppose a friend is passing through town and calls to say I m within half a mile of your house I Question What can be said about where your house is in relation to the friend C 2mm Nancy Pfenning Eiementary Statistics Looking althe Big Picture L21 an Elementary Statistics Looking at the Big Picture Example How Far is One from the Other I Background Suppose a friend is passing through town and calls to say I m within half a mile of your house I Response C 2mm Nancy Pfenning Eiementary Statistics Looking althe Big Picture L21 32 C 2007 Nancy Pfenning De nitions Margin of Error Distance around a sample statistic within which we have reason to believe the corresponding parameter falls A common margin of error is 2 sds Con dence Interval for parameter Interval within which we have reason to believe the parameter falls range of plausible values A common con dence interval is sample statistic plus or minus 2 sds A Closer Look A parameter is not a R V mdoes not obey laws ofprobability ne word eonfidenee 2mm mnwmm amnuwsmsm mm We swim in 33 Example Con dence Interval for p El Background304000075 students picked 7 at random from 1 to 20 Let s assume sample proportion for n400 has sd 001 El Question What can we claim about population proportion p picking 7 2mm mm mm ammw Stalstics mm um aw mm L211 Example Con dence Interval for p El Background30400075 students picked 7 at random from 1 to 20 Let s assume sample proportion for n400 has sd 001 El Response Example Con dence Interval for p Looking Back My probability statement claimed sample proportion should fall within 2 sds of population proportion Now the inference statement claims population proportion should be within 2 sds of sample proportion 2 mm mm mm w as By pretty sure we mean 95 con dent because 95 is the probability of sample proportion within 2 sds of p for large enough 71 2mm mm mm ammw Stalstics mm um aw mm L211 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning 1i i Example con dence Intervalforp Probability Interval for 13 Picking 7 Population proportion is known produce Looking Back In Data Production we learned about probability interval for sample proportion biased samples The data suggest pgt0 05 students were apparently biased in favor of 7 Their selection were haphazard not random If sampling for observational stuajz or assignment to experimental treatments is not random then we produce a con dence interval that is not centered at p population I 05 02 2 standard 95 probability interval for sample proportion picking seven I Sale t W d39pe propor Ion In repea e ran om samples I L2139 o 2mm Nancy Pfenning Elementary Statistics Looking attne Big Picture L21 38 o 2mm Nancy Pfenning 7 Con dence Interval for p Picking 7 Behavior of Sample Proportion Review Measure sample proportion produce confidence For random sample of Size n from population interval for unknown population proportion w1th p 1n category of 1nterest sample We do not sketch a curve showing proportion if has probabilities for population proportion because it is not a random variable I mean 9 l l standard dev1at1on p n p i samp39e pr p quot39 quot 39 We do 1nference because p 1s unknown how one random sample A Closer Look How do we know can we know the standard dev1at1on wh1ch the margin of involves 95 confidence interval for error W Unknown popu atlon proportion L21 Am C 2mm Nancy Prenniiig Elementary Statistics Luuklng attne Big Picture L21 41 o 2mm Nancy Pfenning Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning i J De nition Standard Deviation vs Standard Error Standard error estimated standard We estimate standard deviation of 13 deviation of a sampling distribution with standard error A A p 1 p n Looking Ahead In many situations throughout inference when needed information about the population is unknown we substitute known information about the sample e 2mm Nancy Pfenning Elementary Statistics Luuking attne Big Picture iii 42 e 2mm Nancy Pfenning Eieinentaiy statisties Luuking attne Big Picture iii 43 De nition Con dence Interval Formula Conditions 95 con dence interval for p approx 95 con denpi approx 19 p A 2 151 15 39 n I I Sample must be unbiased otherwise interval is not really centered at 13 I estimate samlple proportion I 11 must be large enough so 23 is approx normal Stanhara for otherwise multiplier 2 from 6895 997 Rule is margin of error incorrect 2 Standard errors I Population size must be at least lOn 95 con dence interval otherwise formula for sd which requires for population proportion independence is incorrect e 2mm Nancy Pfenning Elementary Statistics Luuking attne Big Picture iii 44 e 2mm Nancy Pfenning Elementary Statistics Luuking attne Big Picture iii 45 Elementary Statistics Looking at the Big Picture 9 C 2007 Nancy Pfenning Conditions for Normality in Confidence Interval Example Cheekmg sample Size Multiplier 2 from normal dist approximately I Background 304000075 students picked correct if np and nlp both at least 10 7 at random from 1 to 20 BUtP is unknown 50 sunsnnne 13 3 III Question Do the data satisfy requirement for Require m3 nXn approximate normality of sample proportion Sample count in X and out nX of category of interest should both be at least 10 C 2mm Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Eilg F39lCturE iii 46 C 2mm Nancy F39fErlrllrlg Elementary Statistles Lunklng attne Eilg F39lCturE iii 47 Example Checking Sample Size Example Checking Population Size I Background 304000075 students picked I Background To draw conclusions about 7 at random from 1 to 20 criminal histories of a city s 750 bus drivers a D Response random sample of 100 drivers was used I Question Is there approximate independence in spite of sampling without replacement so formula for standard error is accurate C 2mm Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Eilg F39lCturE iii 43 C 2mm Nancy F39fErlrllrlg Elementary Statistles Luuklng attne Eilg F39lCturE iii an Elementary Statistics Looking at the Big Picture 10 C 2007 Nancy Pfenning E Example Checking Population Size Example Revisiting Original Question I Background To draw conclusions about I Background In sample of 446 college criminal histories of a city s 750 bus drivers a students 246 proportion 055 ate breakfast random sample 0f100 driVerS was used El Question Assuming sample is representative II Response what interval should contain proportion of all students at that university who eat breakfast C 2mm Nancy Ptenning Elementary Statistics tciciving atlne Big Picture mi 52 C 2mm Nancy Ptenning Elementary Statistics Luuking atlne Big Picture mi 53 Example Revisiting Original Question Example Role of Sample Size El Background In sample of 446 college El Background 95 confidence intervals are StUdentS 246 PTOPOI UOH 055 ate breakfa5t constructed based on sample proportion 054 I Response Approx 95 confidence interval from various sample sizes fOI p is A i 2 131 15 sample 95 confidence p n size at standard error 0ft margin of error interval 50 VW 070 2o7 14 4068 200 VW 035 2035 07 47 61 Looking Back Earlier we wondered ifa mayorin of 1000 3954 6554 016 2016 03 BL 57 students eat breakfast I The interval suggests this S D Question What happens as n increases the case Since it IS entirely above 050 cnum Nancy Pfenning Elementary Statistics Luuking attne Big Picture w 56 cnum Nancy Pfenning Elementary Statistics Luuking attne Big Picture w 57 Elementary Statistics Looking at the Big Picture 11 C 2007 Nancy Pfenning Example Role of Sample Size Example Role of Sample Size 0 95 confidence intervals for population proportion I Background 95 A con dence intervals are favoringthe candidate constructed based on sample proportion 054 from various sample sizes quot50 4 0 g margin of 258 sample 95 confidence W Size n standard error ofp margin of error Interval 1200 Cl ggrio k 50 54g63954 070 2o7 14 407 68 47 54margin 6 100 115 2 0721mm 3 39 r 200 54216054 035 2035 07 47 61 Mi Seei pflfIg pal Size wzi39lr margin 1000 016 2016 03 51 57 000 5 5 g V quw 5 M 21323 39 II Res onse lar er n9 39 p g I Response larger n9 02007 Nancy Pfenning Eiementary Statistics L00ilting attne Big Picture LN 59 02007 Nancy Pfenning Eiementary Statistics L00ilting attne Big Picture LZi Bi Other Levels of Con dence Other Levels of Con dence Con dence level 95 uses multiplier 2 Other Con dence level 95 uses multiplier 2 Other levels use other multipliers based on normal curve levels use other multipliers based on normal curve More precise multiplier for 95 is 196 instead of 2 Level Multiplier 90 1645 90 I 95 1960l 95 kg 98 98 2326 0 i t9 gt 99 2 576 i 4 45 0 5i z 39 1960 1960 23 6 2326 2576 2576 02007 Nancy Pfenning Eiementary Statistics L00ilting attne Big Picture LN 62 02007 Nancy Pfenning Eiementary Statistics L00ilting attne Big Picture LN 63 Elementary Statistics Looking at the Big Picture 12 C 2007 Nancy Pfenning J i J Example Other Levels of Con dence Example Other Levels of Con dence I Background Of 108 students in committed relationships 070 said they took comfort by sniffing outoftown partner s clothing Standard error can be found to be 004 I Question How do 90 95 98 99 confidence intervals compare I Background Of 108 students in committed relationships 070 said they took comfort by sniffing outoftown partner s clothing Standard error can be found to be 004 I Response 90 CI is 063 077 95 CI is 062 078 98 CI is 061079 99 CI is 060 080 C 2mm Nancy Ptenning Elementary Statistics Luuking allne Big Picture mi B4 C 2mm Nancy Ptenning Elementary Statistics Luuking allne Big Picture mi BB i i Example Other Levels of Con dence Confidence Interval and LongRun Behavior Intervals get as confidence level increases Repeatedly set up 95 confidence interval for szmgggnee f mgfst proportion of heads based on 20 comflips 5 7 3 In the long run 95 of the intervals should ismvgugm fi contain population proportion of heads 05 0 70 go 98 confidence i l interval lor p 4 60 70 80 highest level 4 99 con dence at confidence interval lor p Wm 60 70 30 C 2mm Nancy Pfenning Elementary Statistics Luuking allne Big Picture mi BE C 2mm Nancy Ptenning Elementary Statistics Luuking allne Big Picture mi BB Elementary Statistics Looking at the Big Picture Confidence Interval and LongRun Behavior 1 J 20 coin flips 95 confidedce interval WHHTHTHTFHH HTH proportion 01 heads 920 45 15 20 2530 354 4 055606570 758055 HTI HHTHHTI I39HTHTTFHHT pom f heads 820 40 15 20 25 30 3545 055 60 65 70 75 80 85 I 15 20253035 4045i06 6 65 70 75 8085 I 1 i I proportion 01 hea05122060 T T T T W Wquot proporiion otheads152075 15 202530 354045 0 55 6065 7069 80 85 repeated 39in the long run I ilips of 20 We oi intervals contain 050 39 coins n of intervals do not contain 50 l lll 1Trr TI39THTTHHTHHH 1 40 15 20 25 303545 Igo 5560 6570 75 8085 TTH H proportion of heads 820 C 2007 Nancy Pfenning Elementary Statistics Luuking althe Big Picture L2170 C 2007 Nancy Pfenning E Example Con dence in the Long Run I Background Presidentelect Barack Obama39s campaign strategists weren39t the only ones vindicated Tuesday Pollsters came out looking pretty good too Of 27 polls of Pennsylvania voters released in the campaign39s nal two weeks only seven missed Obama39s 103point victory by more than their margins of error Obama39s national victory of about 6 points was within the error margins of 16 of the 21 national polls released in the final week I Question Should pollsters be pleased with success C rates of 20271621 75 2007 Nancy Pfenning Elementary Statistics Luuking althe Big Picture L21 71 Example Con dence in the Long Run I Background Presidentelect Barack Obama39s Tuesday Pollsters came out looking pretty good Obama39s 103point victory by more than their national polls released in the final week 1 Response margin of the time C 2007 Nancy Pfenning Elementary Statistics Luuking althe Big Picture campaign strategists weren39t the only ones vindicated too Of 27 polls of Pennsylvania voters released in the campaign39s nal two weeks only seven missed margins of error Obama39s national victory of about 6 points was within the error margins of 16 of the 21 they should come within the error L2173 Elementary Statistics Looking at the Big Picture Lecture Summary Inference for Proportions Con dence Interval El 3 forms of inference focus on con dence interval CI Probability vs con dence I Constructing con dence interval El 0 2007 Nancy Pfenning I Margin of error based on standard error I Conditions Role of sample size Con dence at other levels Con dence interval in the long run Elementary Statistics Luuking althe Big Picture L10 74 Lecture 10 Relationships Two Categorical Variables uTwoWay Tables uSummarizing and Displaying uComparing Proportions or Counts uConfounding Variables 2 mm mm mm Emma sums mm aims Biv more C 2007 Nancy Pfenning Looking Back Review El 4 Stages of Statistics I Data Production discussed in Lectures 14 I Displaying and Summarizing El Single variables 1 cat1 quan discussed Lectures 58 El Relationships between 2 variables Categorical and quantitative discussed in Lecture 9 a Two categorical I Twoquamiralivz l Probability I Statistical Inference 2mm mm mm Elementaiv slums mm mm Biv Vinaquot mu 2 Single Categorical Variables Review El Display I Pie Chart El Summarize Add categorical explanatory variable 9 display and summary of categorical responses are extensions of those used for single categorical variables 2mm marer Elemenhwstahshcs mm aims BivVietuie mu Example Two Single Categorical Variables Elementary Statistics Looking at the Big Picture El Background Data on students gender and lenswear contacts glasses or none in twoway table c G N Total F 121 32 129 282 M 42 37 85 164 T0131 163 69 214 446 El Question What parts of table convey info about the individual variables gender and lenswear 2mm mm mm Elementaiv slums mm mm Biv Vinaquot Lin 4 C 2007 Nancy Pfenning Example Two Single Categorical Variables El Background Data on students gender and lenswear contacts glasses or none in twoway table c G N Total F 121 32 129 282 M 42 37 85 164 D Response Tmal 163 69 214 446 I is about gender I is about lenswear 2mm mmmnm Eiemenhwslahshcg mm alth swim mus Example Relationship between Categorical Variables El Background Data on students gender and 1enswear contacts glasses or none in twoway table c G N Total F 121 32 129 282 M 42 37 85 164 31 163 69 214 446 El Question What part of the table conveys info about the relationship between gender and lenswear mumm mm ammwsmm mmumaw mm m Example Relationship between Categorical Variables El Background Data on students gender and lenswear contacts glasses or none in twoway table c G N Total F 121 32 129 282 M 42 37 85 164 Tmal 163 69 214 446 El Response is about relationship 2mm mmmnm Eiemenhwslahshcg mm alth swim Lag Summarizing and Displaying Categorical Relationships Elementary Statistics Looking at the Big Picture El Identify variables roles explanatory response El Use rows for explanatory columns for response El Compare proportions or percentages in response of interest conditional percentages for various explanatory groups El Display with bar graph l Explanatory groups identi ed on horizontal axis I Conditional percentages or proportions in responses of interest graphed Vertically mumm mm ammwsmm mmmmw mm tam De nition I A conditional percentage or proportion tells the percentage or proportion in the response of interest given that an individual falls in a particular explanatory group C 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture Li a ii C 2007 Nancy Pfenning Example Comparing Counts vs Proportions I Background Data on students gender and lenswear contacts glasses or none in twoway table C G N Total F 121 32 129 282 M 42 37 85 164 Total 163 69 214 446 El Question Since 129 femaes and 85 males wore no lenses should we report that fewer males wore no lenses C 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture Li u 2 i Example Comparing Counts vs Proportions I Background Data on students gender and lenswear contacts glasses or n one in twoway table C G N Total F 121 32 129 282 M 42 37 85 164 Total 163 69 214 446 ii Response C 2mm Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture Li u 14 Elementary Statistics Looking at the Big Picture Example Displaying Categorical Relationship I Background Counts and conditional percentages produced software Rows Gender Columns Lenawear Contacts glasses none A11 1 32 1 female 1 29 282 4291 1135 4574 10000 male 42 37 85 154 2551 2256 5183 10000 All 163 69 214 446 II Question How can we display this information C ZEIEI7 Nancy Ptenning Eiernentary Statistics Luuking attne Big Picture Li u is C 2007 Nancy Pfenning i J E Example Displaying Categorical Relationship Example Interpreting Results I Background Counts and conditional percentages I Background Counts and conditional percentages Rows bender columns Lenswear produced w1th software contacts glasses Au produced w1th software Gender 1mm Lemme female 121 32 129 282 contacts glasses none All 4291 1135 4574 10000 female 121 32 129 282 male 42 37 85 164 25 61 22 4 56 5183 100 00 male 42 a7 85 164 All 163 69 214 446 22456 1003900 El Response A11 163 69 21A 446 I Questions Are you convinced that in general I all females wear contacts more than males do I all males are more likely to wear no lenses Caution Ifwe made lenswear explanatory we d compare 129214 60 with no lensesfemale 85214 40 with no lenses male etc Whv is this not useful MD 17 CJZEIEI7 Nancy Ptennirig L1EI1B Eierneritaiy Statistics Luuking atthe Big Picture Eiementaiy Statistics Luuking atthe Big Picture CZEIEI7 Nancy Ptenning Example Interpreting Results Example Comparing Proportions I Background Counts and conditional percentages I Background An experiment considered if wasp larvae were less likely to attack an embryo if it was a produced with software Gander cums contacts glasses none All female 121 32 129 282 brother 4291 1135 4574 10000 Attacked Not Total male 42 37 85 154 attacked 2561 2256 5183 10000 All 153 69 214 446 BrOther 16 15 31 Unrelated 1 Responses 24 7 31 Total 40 22 62 I Contacts I NO lenses II Question What are the relevant proportions to Looking Ahead Inference will let usjudge ifsample differences are large compare enough to suggest a general trend For now we can guess that thefirst difference is real due to different priorities for importance of appearance i uu iveiic Pieiiiiiiiu LiU i c ZEIEI7 Nancy Ptennirig Eiernentaiy Statistics Luuking atthe Big Picture MD 22 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Comparing Proportions El Background An experiment considered if wasp larvae were less likely to attack an embryo if it was a brother 15 7 22 El Response I Brother iwere attacked I Unrelated 7 Were attacked 7 likely to attack a brother Wasp 6mm mm mm mm aw Mm 2 mm mm mm mm Another Comparison in Considering Categorical Relationships El Instead of considering how different are the proportions in a twoway table we may consider how different the counts are from what we if the explanatory and response variables were in fact unrelated 2mm mm mm ammw mm mm um aw mm ma Example Expected Counts El Background Experiment considered if wasp larvae were less likely to attack embryo if it was a brother 16 15 31 24 7 31 22 62 El Question What counts would we expect to see if being a brother had no effect on likelihood of attack amnuwsmsm mm tithe swim mu27 Example Expected Counts Elementary Statistics Looking at the Big Picture El Background Experiment considered if wasp larvae were El Response 2mm mm mm ammw mm mm um aw mm mm cm C 2007 Nancy Pfenning Example Comparing Counts El Background Tables of observed and expected counts in wasp aggression experiment Exp A NA T 16 15 31 B 20 11 31 24 7 31 U 20 11 31 22 62 T 40 22 62 El Question How do the counts compare 2mm mnwmm amnuwsmsm mm tithe swim mum Example Comparing Counts El Background Tables of observed and expected counts in wasp aggression Obs A NA T B 16 15 31 U 24 7 31 T 40 22 62 El Response 2mm mm mm ammw mm mm um aw mm mm 34 Example Expected Counts in Lenswear Table El Background Data on students gender and lenswear contacts glasses or none in twoway table c G N Total F 121 32 129 282 M 42 37 85 164 T al 163 69 214 446 El Question What counts would we expect to wear glasses if there were no relationship between gender and lenswear 2mm mnwmm amnuwsmsm mm tithe swim mu35 Example Expected Counts in Lenswear Table Elementary Statistics Looking at the Big Picture El Background Data on students gender and lenswear contacts glasses or none in twoway table c G N Total F 121 32 129 282 M 42 37 85 164 T a1 163 69 214 446 El Response Altogether 69446 wore glasses If there were no relationship we d expect females and males with glasses 2mm mm mm ammw mm mm um aw mm mm 37 C 2007 Nancy Pfenning E Example Observed vs Expected Counts Example Observed vs Expected Counts II Backgroundzlf gender and lenswear were unrelated we d expect 44 females and 25 males with glasses C G N Total F 121 32 129 282 F 121 32 129 282 M 42 37 85 164 M 42 37 85 164 Total 163 69 214 446 Total 163 69 214 446 El Backgroundzlf gender and lenswear were unrelated we d expect 44 females and 25 males with glasses C G N Total II Question How different are the observed and El Response Considerably females and expected counts of females and males with glasses males wore glasses compared to what would be expected if there were no relationship C 2mm Nancy Pfenning C 2mm Nancy Pfenning Eiementary Statistics Looking atthe Big Picture LiEISE Eiementary Statistics Luuking atthe Big Picture Li a 4n Confounding Variable in Categorical Example Confounding Variables Relatlonsh1ps II results for fulltime students I If data in twoway table arise from observational study consider possibility of confounding variables Looking Back Sampling and Design issues should always be g m considered before reporting summaries of single variables or D relationships Em u I Question Is there a relationship between whether or not major is decided and living on or off campus C 2mm Nancy Pfenning Eiementary Statistics Looking atthe Big Picture LB 4i C 2mm Nancy Pfenning Eiementary Statistics Looking atthe Big Picture Li a 42 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning i i E Example Confounding Variables Example Handling Confounding Variables II results for fulltime students I Background Year at school may be confounding variable in relationship between major decided or not and living on or off campus El Question How should we handle the data mil Pevcent an or on camaus d cil ef El Response C Zuu7 Nancy Ptenning Eiementary Statistics Luuking attne Big Picture Li D 44 e Zuu7 Nancy Ptenning Eiementary Statistics Luuking attne Big Picture Li D 45 Example Handling Confounding Variables Example Confounding Variables I Background Year at school may be confounding I Background Year at school may be confounding variable in relationship between major decided or variable in relationship between major decided or not and living on or off campus not and living on or off campus El Response Separate according to year 1St and 2nd El Response Separate according to year 1St and 2nd underclassmen or 3rd and 4th upperclassmen underclassmen or 3rd and 4th upperclassmen i Underclassmen i On Campus i Off Campus i Totali Rate On Campus Upperclassmen i On Campus Off Campus i Total il Rate On Campusl i Und c39ded i 117 i 55 i 172 Ii 11717268 o i Undecided i 7 i 26 i 33 i 73321 i Dec39ded i 82 i 37 i 119 Ii 8211969 i Decided i 14 i 92 i 106 i 1410613 For underclassmen proportlons on For upperclassmen proportions on 5 Z campus are Virtually identical for those campus are pretty close for those oe with major decided or undecided with major decided or undecided c2uu7 Nancy Prens r ing39 m er Eiernentaiy Statistics Looking attne Big Picture Liu4B c2uu7 Nancy F39fEri rling Eiernentaiy Statistics Looking attne Big Picture Liu47 Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning Example Confounding Variables Simpson s Paradox ii Background Year at school may be confounding If the nature of a relationship changes depending variable in relationship between major decided or on whether groups are combined or kept not and 11V1ng on or off campus separate we call this phenomenon t d cc a a El Response Separate according to year 1S and 2H Slmpson s Paradox underclassmen or 3rd and 4th upperclassmen i Upperclassmen i On Campus i Off Campus i Total i Rate On Campus i i Undecided i 7 i 26 i 33 i 73321 i i Decided i 14 i 92 i 106 i 1410613 i mum LookingAheml In Part 4 we will show that this difference 21 vs 13 is not statistically significant e mi Nancy Ptenning Eiementary Statistics Looking atthe Big Picture Li a 48 e mi Nancy Ptenning Eiementary Statistics Looking atthe Big Picture Li a 43 Example Handling Confounding Variables Example Handling Confounding Variables ii Background Suppose that boys like Bart tend to El Background Suppose that boys like Bart tend to eat a lot of sugar and they also tend to be eat a lot of sugar and they also tend to be hyperactive Girls like Lisa tend not to eat much hyperactive Girls like Lisa tend not to eat much sugar and they are less likely to be hyperactive sugar and they are less likely to be hyperactive ii Question Why would the data lead to a El Response misperception that sugar causes hyperactivity e mi Nancy Pfenning Eiementary Statistics Looking atthe Big Picture Li a an e ZEIEI7 Nancy Ptenning Eiementary Statistics Looking atthe Big Picture Li a 52 Elementary Statistics Looking at the Big Picture 9 Lecture Summary Categorical Relationships El EIEIEI EIEI Two Way Tables I Individual Variables in margins I Relationship inside table Summarize Compare conditional proportions Display Bar graph Interpreting Results How different are proportions Comparing Observed and Expected Counts Confounding Variables 2mm mnwmm Eiemenuwsuusucs mm tithe swim we Elementary Statistics Looking at the Big Picture C 2007 Nancy Pfenning

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