Test #2 Study Guide!!
Test #2 Study Guide!! PY 211
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This 32 page Study Guide was uploaded by Allie Newman on Thursday October 8, 2015. The Study Guide belongs to PY 211 at University of Alabama - Tuscaloosa taught by Rebecca Allen in Summer 2015. Since its upload, it has received 67 views. For similar materials see Elem Statistical Methods in Psychlogy at University of Alabama - Tuscaloosa.
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PY 211 Test 2 Study Guide chapters 5 6 7 8 Chapter 5 New Statistical Notation pA probability of eventoutcome A H m andquot eg pAmB probability that both A and B are true H H U or eg pAuB probability that eitherA or B are true given thatquot eg pAB probability that A is true if we already know that B is true Chapter 5 l Bayes Theorem puP puP pu pP page 143 in textbook De nitions Fixed event Any event for which the observed outcome is always the same Random event Any event for which observed outcomes can vary Sample space Total number of possible outcomes for any given event denominator for probability Probability The likelihood that a given outcome will occur symbolized by pevent Probability Frequency of times an outcome occurs divided by the total number of possible outcomes 0 Symbolized as p 0 Used to predict any random event 0 Random event any event where the outcomes observed in that event can vary Unnecessary in a xed event 0 Fixed event any event where the outcomes observed in that event is always the same 0 Operational de nition of probability 0 Frequency that a given outcome occurs divided by the total number of possible outcomes 0 See p0 werpoint for equations Calculating Probability To calculate we need to know 0 Number of total possible outcomes or the sample space 0 How often an outcome of interest occurs 0 The equation for probability is where 0 x equals frequency of times outcome occurs 0 f X sample soace pll Probabilities o 1Vary from Oto 1 0 Can be written as a fraction decimal or proportion 0 Larger number greater likelihood of outcome 0 Within the sample space for a given event 0 O O O 1 Outcomes are mutually exclusive amp exhaustive only one thing can happen at a time Sum of probabilities of all possible outcomes 10 something has to happen 50 events are binary they happen or not probabilities are relative likelihoods of those events Probability can never be negative An event is either probable probability near 10 or an event is improbable probability near 0 Probability l Examples 0 Of the 69 people who completed our class survey 34 are in a fraternity or sorority lfl randomly select one person from the class what is the probability that person will be in a fraternitysorority o p Greek 3469 493 or 493 0 Of the 69 people who completed our class survey 20 own a brown or black car lfl randomly select one person from the class what is the probability that person will have a brown or black car 0 p Greek 2069 I 0290 or 29 Probability and Relative Frequency 0 Relative frequency of an event is the probability of its occurrence 0 72 nd relative frequency 1 Distribute the frequencies Sum of frequencies equals the sample space By distributing frequencies you nd sample space denominator 2 Distribute the relative frequencies Relative frequency re ects probability for each outcome in the distribution 0 Recall relative frequency distribution 0 Scores graphed in terms of proportional frequency 0 Graph number of safety complaints by employees at 45 small businesses 108 1718 an Beam 5 m1 mega 11 024 53271 39 a 1120 FE SD 7 me 63 E 0111 54 52 a an 45 53 3 Dam N245 100 0 Relationship Between Multiple Outcomes 0 Four relationships can exist between two outcomes 0 1 Mutually exclusive 0 2 Independent 0 3 Complementary o 4 Conditional Mutually Exclusive Outcomes o Mutually Exclusive when two outcomes cannot occur together 0 0AmBO 0 Where m is the symbol for quotandquot 0 Example in one ip of a coin the event it is not possible to ip a head Outcome 1 and a tail Outcome 2 o Additive Rule the probability of any one of these outcomes occurring is equal to the sum of their individual probabilities 0 Wk B nAgt MB 0 Where U is the symbol for quotorquot Independent Outcomes 0 Independent when the probability of one outcome does not affect the probability of the second outcome 0 Example if we ip a coin two times the event it is possible to ip a head Outcome 1 on the rst ip and a head Outcome 2 on the second ip o Multiplicative Rule the probability that both outcomes occur is equal to the product of their individual probabilities 0 Mm 3 prAgtgtlt MB 0 Independent no relationship between events 0 p PM P P o Occurrence probability of one event does not affect occurrence probability of the other Multiplicative Rule the probability that both outcomes occur is equal to the product of their individual probabilities 0 P PmU MN NW 0 If this isn t true then the events are dependent in some way Complementary Outcomes Complementary sum of probabilities is equal to 100 and outcome is exhaustive of all possible outcomes 0 pA pB 100 0 Example if we ip a coin one time the event the probability of ipping a head Outcome 1 or a tail Outcome 2 is 100 The two outcomes head tail are exhaustive of all possible outcomes for this event Subtracting 1 from probability of one outcome will equal probability of the second outcome Relationship between multiple outcomes loint probability that two outcomes both occur 0 p PmU events PmU sample space 0 eg probability of drawing a red face card KQJ x heartsdiamonds 6 52 115 0 Joint probabilities may be either conditional or independent Conditional Outcomes Conditional the probability of one outcome is dependent on the occurrence of the other outcome 0 Probability of occurrence is changed by the occurrence of the other outcome pPUpU U 0 M P my 0 Example You want to determine the probability of drawing two heart cards from a deck A deck has 52 cards of which 13 are hearts 13 On the rst draw p 5 On the second draw the deck has 51 cards only 12 of which are hearts 12 The probability changes p E 0 Probability of one event A given that another event B occurred 0 Symbolized by l 0 Probability of selecting uninsured mother U who gave birth in a public hospital P pPmU80200040 0 Selecting U given she gave birth in P 11UF580130O62 0 Checkn pUPPP UPP4O6562 Conditional Probabilities and Bayes Theorem Bayes Theorem formula that relates the conditional and marginal unconditional probabilities of two conditional outcomes that occur at random 0 This modi ed formula for conditional probabilities can be used to make inferences concerning parameters in a given population IFUNIU U O a a m o Bayes theorem The classic example 0 1 of women aged 40 who have routine screening mammograms actually have breast cancer 0 80 of women with breast cancer will get positive mammograms o 96 of women without breast cancer will also get positive mammograms o A 40yo woman had a positive mammogram What is the probability that she actually has breast cancer o 0 CANCER 0 NO 0 TOTAL 0 CANCER 0 Test 0 80 o 950 o 1030 Pos 0 Test 0 20 o 8950 o 8970 Neg 0 TOTAL 0 100 o 9900 0 10000 0 8001 103010000 078 0 801073 078 Summarizing Bayes theorem allows us to estimate the conditional probability of a given event based on what we know about the probabilities of each of the events involved 0 Also introduces marginal probability 0 Likelihood of an event independent of other events 0 More on marginal probabilities When we talk about hypothesis testing Probability Distributions Random Variable l a variable obtained or measured in a random experiment 0 It is not the actual outcome of a random experiment but describes the possible outcomes in a random experiment 0 Can describe the outcomes for any behavior that varies from person to person or from situation to situation 0 Probability Distribution distribution of probabilities for each outcome of a random variable 0 Probability for obtaining each possible outcome of a random variable 0 Each probability in a distribution ranges from 0 to 1 and can never be negative 0 Sum of probabilities in a distribution for a random variable X is equal to 1 ZPx100 o Researcher records number of participants rating certain situation as stressful 5point scale Construct probability distribution for ratings The Frequency fX and Relative Frequency pX TABLE 53910 Distribution of Participant Ratings Ratings I x I pm 1 20 250 2 28 350 3 14 175 4 10 125 5 8 100 I ZkZZBO Zpooz LOO TABLE 511 The Probability Distribution of Participant Ratings pX 250 350 175 125 100 I The distribution sums to 100 Summarizing 0 Joint probabilities conditional vs independent 0 Basis for understanding associations among variables 0 One criterion for causation l event P changes likelihood of event U o lnferential statistics using sample characteristics to make probability statements about the population 0 Hypothesis testing quothow likely is it that my data would look like this if there were m association between the variables ie if probabilities were independentquot Mean of a Probability Distribution and Expected Value 0 Expected Value the mean or average expected outcome for a given random variable 0 72 compute the mean value 1 Multiply each possible outcome X times the probability of its occurrence 0 2 Sum each product HZW Expected Outcome the sum of the products for each random outcome times the probability of its occurrence o It is necessary to determine the distribution of all other outcomes for a random variable 0 Expected value gives only the average outcome of a random variable 0 This is determined by computing 0 Variance of a probability distribution 0 Standard deviation of a probability distribution 0 Expected value average outcome of a random variable 0 Also need to describe the distribution of all other outcomes for a random variable 0 Same statistics as with any distribution 0 Variance 0 Standard deviation Expected Value and the Binomial Distribution Binomial Probability Distribution distribution of probabilities for each outcome of a bivariate random variable only 2 possible outcomes 0 Can occur by natural occurrence Example outcomes for ipping a coin are heads or tails No other ways to de ne the outcomes for this random variable 0 Can occur by manipulation Example outcomes for selfesteem among children could be high or low Could de ne as having more outcomes but manipulated data to only have two possible outcomes Mean of a Binomial Distribution 0 The product of the number of times the random variable is observed n times the probability of the outcome of interest on an individual observation p Die7p Homework for Chapter 5 5 6 12 14 18 20 22 24 26 28 30 32 Chapter 6 Normal Distribution 0 The theoretical distribution with data that are symmetrically distributed around the mean median and mode 0 Scores closer to the mean are more probable or likely than scores further from the mean Behavioral data that researchers measure often tend to approximate a normal distribution 0 Normal distribution normally has a mean of O and a standard deviation of 1 Three Examples of a Normal Distribution With Different Means and Standard Deviations 04 02 00 Scores x Frequency of Scares 6202 u0 6210 u0 6205u 2O Characteristics of Normal Distribution 0 1 Normal distribution is mathematically de ned 0 The shape of a normal distribution is speci ed by an equation relating each score along Xaxis with each frequency along y axis 0 Remember rarely do behavioral data fall exactly within limits of above formula 0 2 Normal distribution is theoretical o Behavioral data typically approximates a normal distribution 0 3 Mean median and mode are all located at the 50th percentile 0 Half of the data 50 in a normal distribution fall above the mean median and mode and half 50 fall below mean median and mode are all at the 50th percentile 4 Normal distribution is symmetrical 0 Distribution of data above the mean is exactly the same as below the mean 0 Folding test fold distribution in half perfect overlap 5 The mean can equal any value 0 Mean can equal any number from positive in nity 00 to negative in nity oosMsoo 6 The standard deviation can equal any positive value 0 Data can vary SDgtO or not vary SDO o Describing variability as negative is meaningless 7 Total area under the curve is equal to 10 0 Area under the curve varies between 0 and 1 and can never be negative relative frequency 0 Proportions of the area are used to determine the probabilities for normally distributed data 0 8 The tails of a normal distribution are asymptotic o Tails of the distribution are always approaching Xaxis but never touch it o Allows for possibility of outliers in a data set Remember the empirical rule 0 It s true of every normal distribution 0 But if we convert it to a standard normal distribution we can begin to assign real meaning to the scores along the distribution Standard Normal Distribution 0 A normal distribution with mean O and standard deviation 1 0 Given this knowing any given value tells you exactly where it falls on the distribution 0 Any normal distribution can be standardized by computing the zscore 0 Value on the xaxis of a standard normal distribution 0 Speci es distance in SDs of a value from the mean o Yaxis is relative frequency proportion or probability Research in Focus lThe Statistical Norm Researchers use the word normal to describe behavior 0 Example researchers studying links between obesity and sleep duration state that short sleepers are at higher risk of obesity compared to normalsleepers o What does it mean when researchers say normalsleepers 0 Does the sleep pattern in a population follow a normal pattern What is normal m The Statistical Norm Only 5 of data fall in the tails beyond 2 SD from the mean these data are not normal or likely 95 of data fall within 2 SD of the mean these data are normal or likely Behavioral data that fall within 2 SD of the mean are regarded as normal or likely because these data fall near the mean Behavioral data that fall outside of 2 8D from the mean are regarded as not normal or not likely because these data fall far from the mean in a normal distribution Standard Normal Distribution 0 Standard Normal Distribution a normal distribution with a mean equal to O and a standard deviation equal to 1 0 Distributed in zscore units along the Xaxis zScore l value on the Xaxis of a standard normal distribution The numerical value specifies the distance or standard deviation of a value from the mean 0 Standard Normal Transformation 0 quotnormalizing 0r quotstandardizingquot the scores 0 Converts any normal distribution to a standard normal distribution M O and SD 1 Transforms raw scores to zscores Pinpoints each score in relation to the mean 0 Distance Direction 0 Standard Normal Transformation H a formula used to convert any normal distribution to a standard normal distribution with a mean of O and standard deviation of 1 X 2 G X M SD for a population of scores for a sample of scores 0 Use transformation to locate where a score would fall in the standard normal distribution 0 Once you know location standard normal distribution rules to nd the probability Example 61 Standard Normal Distribution 0 Relative positionfrequency doesn t change The data were normally distributed with M 12 and SD 2 What is the zscore for X 0 14 feet lb Standard illlairmal Distribution El riigiinal Distributinn 431 1 a 1 3 l a 1h li i x1 eis zzlsn above the mean ahare the mean 0 Because M 12 and SD 2 we can nd the zscore for X 14 by substituting these values into the ztransformation 14 1 2 100 o The Unit Normal Table 0 Appendix B of your text Probability distribution table displaying 0 A list of zscores and o The corresponding probabilities or proportions of area associated with each 2 score listed The unit normal table has three columns A B and C TABLE 61 A Portion of the Unit Normal Table in B1 in Appendix B A B 1 000 0000 5000 001 0040 4960 002 0080 4920 003 0120 4880 004 0160 4840 005 0199 4801 006 0239 4761 007 0279 4721 008 0319 4681 009 0359 4641 010 0398 4602 011 0438 4562 012 0478 4522 013 0517 4483 014 0557 4443 Source Based on J E Freund Modern Elementary Statistics 1 1th edition Pearson Prentice Hall 2004 0 Column A Lists the zscores 0 Only lists positive zscores for negative 2 scores recall that distribution is symmetrical 0 Listed from z O at the mean to z 400 above the mean 0 Column B Lists the area between a zscore and the mean 0 First value is 0000 area between the mean and Z O 0 As 2 score moves away from mean proportion of area between score and mean increases closer to 50000 0 Column C Lists the area from a zscore toward the tail 0 As 2 score increases area between that score and tail decreases closer to 0000 Locating Proportions Area at each zscore is given as a proportion in the unit normal table 0 Can use the unit normal table to locate the proportion or probability for a score 0 To locate the proportion 0 Step 1 Transform a raw score X into a zscore 0 Step 2 Locate the corresponding proportion for the zscore in the unit normal table Example 67 Locating Proportions Between Two Values 0 Data indicative of chewing gum improving academic performance by mean scores on a standardized math test of 20 i 9 points Assuming these data are normally distributed what is the probability that a student who chews gum will score between 11 and 29 points higher the second time he or she takes the test GURE Hill n A Normal Distribution Wth M 2 rm and SD 9 3413 341 3 The shaded region is the proportion of ecoree between X 1 1 and Jr 99 in this dlEllIl39TbutlDIl39l Step 1 forX 11 0 To transform raw score x to a z score compute a z transformation 2 1120 9 1oo 9 9 0 Step 2 forX 11 0 Find the zscore 100 in Column A of Table Bl in Appendix B then look in Column B for the proportions between 100 and the mean 0 3413 0 To nd proportion between two scores add 3413 to the proportion associated with the second score X 29 0 Step 1 forX 29 0 Compute the ztransformation for X 29 z 2922100 9 9 0 Step 2 forX 29 o The proportion between the mean and a zscore of 100 is the same as that for 100 p 3413 The total proportion between 11 and 29 is the sum of the proportion for each score 0 34133413 6826 Locating proportions 0 So transformation to zscore allowed us to estimate proportion of scores that are greater than 12 in the raw distribution 0 p2 gt 2 0228 I px gt 12 0228 1 9 3 Itz 3 D 2 4 6 a 10 12 14 Locating Scores 0 The unit normal table can be used to locate scores that fall within a given proportion or percentile o 72 locate the score 0 Step 1 Locate a zscore associated with a given proportion in the unit normal table 0 Step 2 Transform the zscore into a raw score X Example 68 Locating Scores 0 In the general healthy population IQ scores are normally distributed with 100i15uiro AssumIng these data are normally distributed what IS the minimum score needed to be in the top 10 of this distribution of intelligence scores in this distribution FIIEIIJFEE E1l1l Locating Spares for Example 58 l Scares Standard Normal Distribution Distribution L uk up in ztablef 339 123 Use ztransfurmatiun to solve fnrx Step 1 o The top 10 of scores is the same as p 1000 toward the tail 0 To locate the zscore associated with this proportion look for p 1000 in Column C of the unit normal table in Table Bl in Appendix B o The zscore is Z 128 o A zscore equal to 128 is the cutoff for the top 10 of data Step 2 0 Determine which score X in the distribution shown in Figure 611 corresponds to a z score equal to 128 Since 2 128 you can substitute this value into the z transformation formula 128 X 100 15 0 Multiply both sides of the equation times 15 to eliminate the fraction X 100 I 15128 15 15 192 X 100 0 To nd the solution for X add 100 to each side of the equation 1192 2 X o A score equal to 1192 on the IQ test is the cutoff for the lowest score in the top 10 of scores in this distribution Normal distributions Summary 0 Normal distributions are 0 Theoretical o Mathematically de ned o Symmetrical o Asymptotic Standard normal distribution transformation to make scores immediately interpretable x M Z SD 0 M 0 SD 1 0 Each score transformed to zscore Area under the normal curve 10 total 0 So can use zscore to estimate l by referring the zscore to the unit normal table 0 Proportion of scores falling abovebelow that point 0 Probability of seeing scores abovebelow that point 0 Proportions of scores falling between two values etc 0 Higher absolute value 2 lower probability of occurrence Homework for Chapter 6 Factual problems 8 Concept and application problems 12 15 18 20 25 Problems in research 30 32 Chapter 7 Some new statistical notationssymbols Z zscore standardized score pm mean of the sampling distribution 6M2 variance of the sampling distribution 6 M or SEM standard error of the mean SD of the sampling distribution The Research Process 0 Draw sample of observations from population 0 Use sample characteristics to estimate population parameters lnferential statistics address probability that observations are chance rather than true representation of population Validity of conclusions depends on how well the sample represents the larger population 0 Representative sampling is the backbone of good research Necessarily an imperfect process 0 Affected by 0 sampling design 0 distribution of variable X in the population 0 size of sample drawn 0 Every sample Will differ from every other lnferential Statistics and Sampling Distributions You may ask a few students sample how they scored on an exam compared to your score You do this to learn more about how you did in the entire class the population and not just compared to only those few students the sample FIEUEE iii Selecting Sampler Firen Piiiieuleiiwene ak a queeiiun alequ e1 FUPUIHHUH Example Haw gagd is yciur enure f i an exam egmpared te the mean performance in yeur eleee lF F39LllL TI39DIH lei entity all indieiduele el interest when wane identili ed by the queetien Example All emudents whe eelt the exam in ileum eleee Gbaeruativpna made in the 55m ple are generalized tn the pepulatien Example feu eernieare FEM grade ten the mean grade in the eem ple te determine hijwwell yuan did in yeur eleee Ilihe pe puleiienj EAJMF LE Seleet a partied of individuals frem the pepuiletiien and meeeme a sample alalliatig identi ed HIE ljllJIEJSIEiDI39L Exemple Eeleet a few eludeni e in yeur eleee and needrd their grades en the exam wnn ur 0 Sampling Distributions a distribution of all possible sample means or variances that could be obtained in samples of a given size from the same population 0 You can then compare the statistics obtained in the samples to the value of the mean and variance in the hypothetical population Sampling and Conditional Probabilities To avoid bias researchers use a random procedure to select a sample from a population random sampling 0 To be selected at random all individuals in a population must have an equal chance of being selected 0 The probability of selecting each participant must be the same 0 Sampling Without Replacement when sampling each participant selected is not replaced before the next selection 0 Most common method used in behavioral research 0 Example suppose you place 8 squares on a desk in front of you with two marked A two marked B two marked C and two marked D First draw Probability of selecting a square marked A 325 psquare marked A on 1St draw 8 Second draw Do not replace square A therefore probability of selecting a square marked A 1 pSquare marked A on 2nOI draw 7quot14 0 Sampling With Replacement sampling in which each participant selected is replaced before the next selection 0 Ensures probability for each selection is the same 0 Typically not necessary in behavioral research because the populations are large Example given a population of 100 women 01 Probability of selecting rst woman is p 100 0 Probability of selecting second woman without replacement is p 1 01 99 Selecting a Sample 0 Sample Design speci c plan or protocol for how individuals will be selected or sampled from a population of interest 0 Must address the following two questions Does the order of selecting participants matter Do we replace each selection before the next draw Answering both questions leads to two strategies for sampling 0 Theoretical sampling 0 Experimental sampling The Effect of Replapemenf and Order Changes on the FIGURE 12 Many Possible Samples That Can Be Drawn From 3 Gwen Population I Population N 4 A B c D Theore1ical SampIng Samping SHEERquot Strategies for taking samples used in development of size 2 n irnmtris 0 Slatlgi39 al hfNJ populalion of size 4 N 4 U murdered AAGA NB GB lr l 39 BE 9g Sampling singling mos39i BC 9 coiimonl39y39 used in BD DD ezperirnenial regainsh Sampling Wilh epla emen l Tm fa ms hfe 3mm Ii 0 Sampling 39Will mul R plan In 1 Tmfal possible sample5 1 392 Sampling Strategy lThe Basis for Statistical Theory 0 In theoretical sampling the orderof selecting individuals matters and each individual selected is replaced before sampling again 0 To determine the number of samples of any size that can be selected from a population 0 Total number of samples possible N11 0 Example if we had samples of two participants 7 2 from a population of 3 people N 3 Nquot 32 9 samples Sampling Strategy Most Used in Behavioral Research 0 In experimental sampling the orderof selecting individuals does not matterand each individual selected is not replaced before selecting again 0 To determine the number of samples of any size that can be selected from a population N 0 Total number of samples possible WV 7 0 Example if we select as many samples of two participants as possible 7 2 from a population of 3 people N 3 N 3 3gtlt2gtlt13 nN n 23 2 2gtltIgtlt1 Sampling Distributions of the Mean The The To see how well a sample mean estimates the value for a population mean construct a sampling distribution The sample mean is related to the population mean in three ways 0 The sample mean is an unbiased estimator o It follows central limit theorem 0 It has a minimum variance Distribution of all possible sample means that could be obtained from a population in samples of a given size 0 Population 50 Sample 10 a N39 n N n 10272278170 possible unique samples and that is without replacementll Sampling distribution distributes these as the expected value probability of the mean for each of the 108 possible samples or variances a sampling distribution of the variance Sampling individuals vs sampling values 0 Recall logic of Bessel s correction 0 Less frequent values less likely to be sampled 0 So though each individual has equal pselection each value of the variable does not Sampling distribution displays the probability that each mean value will be observed in a sample of n observations drawn randomly from a population Mean Unbiased Estimator Suppose we select samples of size 2 from a population of 3 with scores of 2 5 and 8 M 5 The sample mean is an unbiased estimator if MZX n 0 When then M p on average Mean Central Limit Theorem Regardless of the distribution of scores in a population the sampling distribution of sample means selected from that population will be approximately normally distributed 0 At least 95 of possible Ms one could select fall within 2 SDs of p Empirical Rule Homework for Chapter 7 Factual problems 1 6 Concept and application problems 20 22 26 Problems in research 28 29 30 31 32 Central Limit Theorem 0 For any population with a mean of u and a standard deviation of 0 the distribution of sample means for sample size n will approach a normal distribution with a mean of M and a standard deviation of 0 0M 0 square root n As n approaches in nity Sampling distribution of the mean a An example A random variable takes 4 possible values 1234 Given an equal poccurrence consider all possible outcomes for a sample n 2 0 11 31 0 12 32 0 13 33 0 14 34 0 21 41 0 22 42 0 23 43 0 24 44 0 Now compute the mean for each of those samples 0 10 20 o 15 25 o 20 30 o 25 35 o 15 25 o 20 30 o 25 35 o 30 40 0 Remember Nn 42 16 possible samples 0 Means n 2 0 Now distribute those the means as a relative frequency probability distribution 13 39 025 02 015 01 005 1 0 o This is the sampling distribution of means Variability of the sampling distribution o The variance of the sampling distributions of the sample means is minimal 0 The standard deviation of sample means is measured by the standard error of the mean 0 To compute the standard error we take the square root of the variance 2 G GM O M 62 MuM2 0 Where M Nquot l number of possible samples of size n The Standard Error of the Mean Variance of a sampling distribution of sample means 2 2 5 GM 0 I 0 Standard error of a sampling distribution of sample means 2 G G M n 0 Sampling Error the extent to which sample means selected from the same population differ from one another 0 The discrepancy or amount of error between a sample statistic and its corresponding population parameter The Standard Error of the Mean 0 Example HA random sample of 100 college students population are asked to state the number of hours they spend studying during nals week The mean is 20 hoursweek and the standard deviation is 15 hoursweek 0 To construct the sampling distribution you must 1 identify the mean of the sampling distribution 2 compute the standard error of the mean 3 distribute the possible sample means 3 SEM above and below the mean 0 1 Identify the mean 0 The sample mean is equal to the population mean so the mean of this sampling distribution is 20 0 2 compute standard error 0 The standard error is the population standard deviation divided by the square root of the sample size 0 GM i 15 150 xE V100 0 3 Distribute possible sample means as shown in Figure 76 Tho Samplian Distribution of Sample Moons llor Samples of FIGURE WE Size ll 141 Selected From a Population With a Moon oil 2quot and Standard Deviation of l 5 loo 15in 1315 o39o oils 2920 oils Sample Means Factors That Decrease Standard Error 0 As the population standard deviation decreases standard error decreases o This is shown in Figure 77 using samples of size 2 n 2 selected from one of ve populations having a population with 61 4 62 9 G3 16 64 25 and 65 81 FIGURE quotIt Population Standard Donation and SEMI i l 35 2 a 533 a mcm ll I I il 9 ll 395 25 339 Population Standard Deviation c As sample size increases standard error decreases 0 Law of large numbers increasing the number of observations or sample size will decrease standard error The smaller the standard error the closer a distribution of sample means will be from the population mean 0 This is illustrated in Figure 78 using samples of size 4 n 4 9 n 9 16 n 16 25 n 25 and 81 n 81 selected from a population with o 4 Fill ulFIE 33 i The Law of Large Humboro 3 2 2 EH 39 too 1 Hugh LED 39339 I ll lI a o is 25 Eli Sample Size in Standard error of the mean 0 Obviously affected by n o largersample size a lower estimated variability 0 Example with n 4 and o 3 l 32 15 o with n 16 and o 3 l 34 075 0 So as n increases 0 likelihood of error in estimating the population mean decreases o sampling distribution begins more amp more to resemble the population distribution Effects of Sample Size 0 Law of large numbers 0 As n increases SE of the mean decreases and hence likelihood that W nu also increases 0 Central Limit Theorem 0 For any population with mean u and standard deviation 0 the distribution of sample means will have a mean u and SD on andl Wilapproximate a normal distribution as n increases 0 Regardless of the distribution of raw values in the population the sampling distribution of the mean Will always approach normality especially with larger n s With respect to the sampling distribution of the mean as sample size n increases the distribution becomes more normal in form Standard Normal Transformations With Sampling Distributions A sampling distribution can be converted to a standard normal distribution by applying the ztransformation Z w Z 0 GM or G M ztransformation is used to determine the likelihood of measuring a particular sample mean from a population with a given mean and variance 0 To locate the proportion and therefore probability of obtaining a sample mean 0 Step 1 Transform a sample mean M into a zscore 0 Step 2 Locate the corresponding proportion for the zscore in the unit normal table Mu Summarizing We have now discussed three types of means 0 1 population mean u o 2 sample mean M o 3 mean of the distribution of sample means law 0 We have also discussed three types for measures of variability o 1 population variance 62 standard deviation 6 o 2 sample variance 52 standard deviation 5 or SD 0 3 variance of the sampling distribution of means 6M2 and standard error of the mean SEM Sampling distribution of the mean displays all possible means for sample size n drawn randomly from the population 0 Mean is unbiased estimator of population mean 0 Variance and thus SEM is minimum possible 0 Sample size plays important role 0 Central limit theorem as n distribution becomes more normal 0 Law of large numbers as n SEM decreases and ppM 1 increases The formula for standard error is either 1 The square root of variance sample size OR 2 The Standard Deviation the square root of the sample size Variance i5 squared Chapter 8 Learning Objectives Identify the SX steps of hypothesis testing Distinguish between the null hypothesis and the alternative hypothesis Explain the concept of statistical signi cance discuss signi cance levels and p values Compute and interpret a onesample ztest De ne Type I error and Type II error and identify how researchers control these errors Explain statistical power and effect size Reportsummarize results in APA format New Statistical Notation d alpha level of signi cance decision criterion pType I error or false positive 3 p Type II error or false negative Ho null hypothesis assumes no effect H1 alternative hypothesis signi cant effect or other number 0 what you think is going to happen p signi cance level pchance difference Hypothesis Testing 0 De nition 0 A method for testing a proposition about a population and its parameters using data measured in a sample drawn from that population oPurpose 0 To test ideas about a population by comparing competing statements about its characteristics LM 0 To determine the likelihood that a given population parameter is true given what we observe in a sample drawn from that population Does our sample look like what we expected What we know to be true Or not 0 Null hypothesis 0 This hypothesis is based on the presumption that our sample means are NOT different 0 THE STRAW MANquot 0 Alternative hypothesis 0 This is the hypothesis that we accept if our procedures indicate that we should reject the null hypothesis o It states that the means are different What we WANTand are actually interested in Our mean is different than the population thus we reject the null hypothesis Start With A Hypothesis o What is a hypothesis 0 A tentative explanation for an observation phenomenon or scienti c problem that can be tested by further investigation 0 What isn ta hypothesis g Fact l something known indisputably to be true 6 Theory organizing conceptual framework that uses broad general principles to describe and explain a phenomenon Within a theory you mightcould have a hypothesis 0 Hypotheses test speci c statements that should be true if the general theory is correct 0 Theories are never proven only supported by data Example 0 At age 2 the mean weight L1 for preterm children is population 0 t126 pounds 0 The distribution is normal with a 04 standard deviation O o The parents of 16 preterm infants are trained to provide their infants increased stimulation through handling music therapy 0 At age two all 16 children are weighed 0 264 22 and 264 30 Expect the 2 year olds to be between 22 to 30 pounds o If these children weigh noticeably more than the general population of preterm two year ods we can assume that touch music therapy aids development 0 Conversely if these children weigh around 26 pounds we can conclude that touch music therapy does not aid development 0 We want our babies to be above 30 Steps of Hypothesis Testing 0 Step 1 State the null and alternative hypothesis IN WORDS o Null Hypothesis H0 statement about the population parameter such as the mean that that there is m effect of interest regarding the population parameter Starting point to determine if null is likely to be true or not EXAMPLE Therapy does not have an effect on weight of preterm babies at age 2 0 Alternative hypothesis H1 statement that contradicts the null hypothesis statement that there is some effect of interest regarding the population parameter EXAMPLE Therapy HAS an effect on weight of preterm babies at age 2 or increases the weight of In any case can predict H1 to be lt gt or H0 0 Step 2 State the null and alternative hypothesis statistically 0 Statement about the population parameter such as the mean that that there is m effect of interest regarding the population parameter Example H0 26 or H0 5 26 Usually 0 but here we know the population mean is 26 lbs 0 Alternative hypothesis H1 statement that contradicts the null hypothesis statement that there is some effect of interest regarding the population parameter Example H1 26 or H13 26 Usually 0 but here we know the population mean is 26 lbs 0 Step 3 Set the criteria for a decision l pg 239 in book good table to look at 0 Done by stating the level of signi cance Criterion ofjudgment upon which a decision is made regarding the value stated in a null hypothesis 0 HOW sure do you need to be before you are Willing to reject the null hypothesis alpha level a decision criterion level of certainty margin of error 0 Probability value that de nes maximum level of chance you will accept in choosing H1 over H0 0 Inferentia statistics a probability statements about correctness of alternative hypothesis 0 Use the null hypothesis to predict what kind of sample mean we can expect to obtain if treatment didn t work We can also predict which sample means are consistent with the null hypothesis and which ones are inconsistent with the null hypothesis ie consistent with alternative hypothesis 0 We can split the sample means into two groups 0 Those that are likely to be obtained if the null hypothesis is true 0 Those that are unlikely if the null hypothesis is true 0 Step 3 Continued I Criterion for making a decision 0 Traditionally we set a 05 Why 5 0 It s conventional 1 chance in 20 low likelihood 0 Empirical rule 5 matches quottailsquot of normal distribution beyond 2 SD Well actually it s 196 ii El 12 is 21 4 3 SD e355 4155 M MED 22 insist quot quot I 68 i 95 9 9 Ho 0 Step 4 Compute the test statistic o The value of test statistic can be used to make a decision regarding null hypothesis Helps determine how likely the sample outcome is if the population mean stated in the null is true 0 The larger the value of the test statistic the further a sample mean deviates from the population mean stated in null hypothesis and ergo that H1 is true Choice of test statistic depends on 0 Scale of measurement of variables 0 Research design 0 Speci c hypothesis being tested 0 Calculate your test statistic zscore in this case Suppose our sample n 16 of 2 year olds weighed M281 pounds m 26 s 4 With d05 is that enough of a difference to suggest that the treatment had an effect M li Z 0 1 039 i 4 100 M M J2 JR 4 39 0 J 0M 100 100 0 Step 5 Make a decision 0 Based on the probability of obtaining a sample mean given that the value stated in the null is true represented by pvalue o 1 Rejectthe null hypothesis l the sample mean is associated with low probability of occurrence if the null is true pvalue lt05 reached signi cance This is what the researcher WANTS o 2 Retain the null hypothesis l the sample mean is associated with high probability of occurrence when null is true pvalue gt05 failed to reach signi cance The straw man is retained The researcher does not want this Example REJECT H0 0 Criterion for making a decision p value obtained results visavis H0 What is the likelihood we would obtain this sample statistic by pure chance given that the true population parameter is this value Smaller p lower Pchance difference if p s a reject Ho accept H1 0 Step 6 State your conclusion about the null and alternative hypothesis IN WORDS o Null Hypothesis Ho I statement about the population parameter such as the mean that is assumed to be true 0 Alternative hypothesis H1 I statement that contradicts the null hypothesis 0 The treatment was effective touch music therapy with the infants helped them gain weight more rapidly and then they weighed more than the general population of preterm 2year0ds on the test extra credit will be this z 21 p is less than 005 Hypothesis Testing 6step process 0 1 State the hypothesis in words 0 H0 The treatment has no effect 0 H1 The treatment has an effect o 2 State the hypothesis statistically 2tailed EX 0 H0 u O 0 H1 I 1 0 o 3 Set the criterion test statistic for decisionmaking o a level usually lt 05 z criterion i196 4 Compute the appropriate test statistic zscore so far 0 5 Evaluate results by comparing the obtained test statistic in Step 4 to the critical value in Step 3 and make a decision 0 pvalue g d a reject H0 pvalue Z d a retain H0 6 State the conclusion in words Hypothesis Testing and Sampling Distributions To locate the probability of obtaining a sample mean in a sampling distribution we must know 0 1 The population mean 0 2 The standard error of the mean The Sampling Distribution for a Population With a Mean Equal to 150 We expect the sample mean to be equal to the population mean If 150 is the correct population mean then the sample mean will equal 1 50 on average with outcomes fartherlrom the population mean being less and less likely to occur Making a Decision and Error 0 Step 5 requires making a decision on Whether to retain or reject the null 0 There are four possible outcomes 1 Decision to retain the null is correct 2 Decision to retain the null is incorrect Type II or 3 error 3 Decision to reject the null is correct 4 Decision to reject the null is incorrect Type I error o In two of the above scenarios an error is committed Types of Error Occurs when conclusion to retain or reject the null hypothesis is incorrect 0 Type and Type II Errors TABLE 83 Four Outcomes for Making a Decision Decision Retain the Null Hypothesis Reject the Null Hypothesis Truth in the True CORRECT TYPE I ERROR Population 1 Oi Oi False TYPE ERROR CORRECT B 1 B POWER The decision can be either correct correctly reject or retain the null hypothesis or wrong incor rectly reject or retain the null hypothesis Type I and Type II Errors 0 Errors in hypothesis testing 0 Type 1 error false positive The probability of rejecting the null hypothesis when it is true Saying our means are different when they are not this is equal to alpha 0 Type 2 error false negative The probability of retaining the null hypothesis when it is not true Saying our means are not different when they are this is equal to Beta Alpha and Power 0 Alpha level Ol level of signi cance or criterion of a hypothesis test 0 Researchers control for Type I error by stating a level of signi cance also called an alpha level 0 The signi cance level is the largest probability of committing Type I error that we will allow and still decide to reject the null hypothesis 0 or level is compared to the pvalue in making a decision 0 Criterion test statistic is that occurring at the locations under the curve where p 05 0 Power the probability of rejecting a false null hypothesis 0 It is the likelihood that we will detect an effect assuming an effect exists Clicker Questions 0 A research report summarizes the results of the hypothesis test by stating z 213 p lt 05quot According to this report 0 The null hypothesis was rejected and the probability of a 73pe error is less than 05 o In terms of the size of the zscore needed for signi cance which is more stringent a onetailed or twotailed test 0 Twotailed less than lt means reject greater than gt means retain Example 81 l NonDirectional TwoTailed Tests 0 Population mean on IQ scores for healthy individuals was IOOiISmiG Does pre school have an effect on IQ 0 Sample of 100 adult participants quot2100 who had attended preschool take the test and obtain a sample mean of 103 quot4103 0 Compute onesample ztest for whether or not we will reject the population mean 2100stated by the null 0505 at a 05 level of signi cance Step 1 State null and alternative hypothesis in words 0 H0 Attending preschool does not have an effect on adult IQ H1 Attending preschool HAS an effect on adult IQ Step 2 State null and alternative hypothesis statistically H0u100 lepsthO Step 3 Set the criteria for a decision 0 The level of signi cance is 05 which makes the alpha level a 05 ln twotailed test divide alpha value in half 0250 in each tail 0 Locate critical values in Table Bl in Appendix B and look up proportion 0250 toward tail in Column C o This value is listed for a zscore equal to i 196 Regions beyond critical values are called rejection region Step 4 Compute test statistic Compute 2 statistic to get obtained value 0 Formula is the sample mean minus the population mean stated in the null hypothesis divided by the standard error of the mean M u o obt 6 GM T o 2 statistic M where n 0 Compute standard error which is the denominator for the 2 statistic o 15 15 G 2 2 0 M 7 100 0 Substitute values for sample mean M103 the population mean stated by the null hypothesis F100 and the standard error we just calculated 2 Z M u 2103 100 GM15 0 5M 15 What decision do you make When you compare the obtained test statistic in Step 4 to the critical z score from Step 3 Reject the null hypothesis 0 ln shaded area always reject if nonshaded area you retain 200 0 Step 5 Make a decision Compare obtained test statistic from Step 4 to the critical value of the z score set in Step 3 o Reject null if obtained value exceeds critical value Obtained value 2011 200 is greater than critical value 2 i 196 and falls in the rejection region 0 Decision in this case is to reject the null hypothesis 0 Step 6 State your conclusion in words Attending preschool has an effect on adult IQ z 200 p lt 05 m Making a Decision for Example 81 The obtained value is 200 which falls in the rejection region 5 reject the null hypothesis 39 Rejection region Rejection region or 0250 OL 0250 Retain the null hypothesis L 3 39 2 39 1 39 a 39 139 39 2 39 3 Null V 1 96 200 1 96 Because the obtained value falls in the rejection region it is beyond the critical value in the upper tail we decide to reject the null hypothesis Example 82 l Directional UpperTail Critical Test 1214mm 0 Reading pro ciency in elementary children increases when children are enrolled in a points reading program 0 Select sample of 25 students enrolled in reading program 7 25 Sample mean equal to 14M14 points improvement 0 Compute onesample ztest at a 05 level of significance oz 05 Step 1 State the null and alternative hypothesis in words 0 H0 The reading program worsens or does not change reading pro ciency in elementary children 0 H1 The reading program improves reading pro ciency in elementary children Step 2 State null and alternative hypothesis statistically H0 312 with the reading program mean improvement is at most 12 points in general population H1 gt12 with the reading program mean improvement is greater than 12 points in general population Step 3 Set the criteria for a decision 0 The level of signi cance is 05 which makes the alpha level 05205 0 To locate critical value locate probability of 0500 toward the tail in Column C o The zscore associated with this probability is z 165 o 165 is the critical value and cutoff for the rejection region Step 4 Compute test statistic Compute zstatistic to get obtained value Z M H o obt 6 GM 2 T o zstatistic M where 7 0 Compute standard error which is the denominator for the zstatistic i080 o G 0 M 0 JE Substitute values for sample mean M 14 the population mean stated by the null hypothesis 12 and the standard error we just calculated GM 03980 M u 103 100 20th 5 080 0 M Step 5 Make a decision 0 Compare obtained test statistic in Step 4 to critical value in Step 3 2075250 2 250 Obtained value is greater than critical value 2 165 and falls in the rejection region 0 Decision is to reject the null hypothesis 0 Since pvalue is less than 5 reject the null hypothesis FIGURE 87 Making a Decision for Example 82 The test statistic reaches the rejection region reject the null hypothesis Rejection region Retain the null hypothesis I I l l I I I 3 2 1 0 1 2 I 3 Because the obtained value falls in the rejection region it is beyond the critical value of 1645 we 0 decide to reject the null hypothesis Step 6 State your conclusion in words 0 The reading program improves reading pro ciency Z 250 p lt 05 One vs TwoTailed Tests OneTailed Tests 0 Greater power 0 If value stated in null hypothesis is false this test will make it easier to detect and reject But dif cult to justify TwoTailed Tests 0 More conservative o More dif cult to reject null hypothesis 0 Most studies in behavioral research are twotailed tests Effect and Effect Size 0 Effect difference between sample mean and population mean stated in null hypothesis 0 Insigni cant when null is retained 0 Signi cant when null is rejected 0 Effect Size H size of an effect in a population 0 How far scores have shifted in the population 0 Percent of variance that can be explained by a given variable 0 Most meaningfully reported with signi cant effects decision to reject null Cohen s d 0 Measures the number of standard deviations an effect shifted above or below the population mean stated by the null hypothesis 0 Value for Cohen s dis 0 when there is no difference between the 2 means and increases as differences get larger 0 Formula M d 0 Cohen 5 6 TABLE 86 Cohen s Effect Size Conventions Description of Effect EffectSize 11 Small dlt 02 Medium 02 lt dlt 08 Large dgt 08 Example 83 l Cohen s d 0 Determining Cohen s dfor Example 82 0 Population mean reading pro ciency improvement was 12i4 6 points In each example the mean test score in our sample was 14 M 14 points improvement What is the effect size for this test using Cohen s 0 o The numerator for Cohen s dis the difference between the sample mean M 14 and the population mean 212 The denominator is the population standard deviation G 4 M 14 12 039 O o 0 Observed effect shifted 050 standard deviations above the mean in the population 0 Students in the elite program score 050 standard deviations higher on average than students in the general population 050 Power and Factors That In uence Power Detecting an effect or power is critical in research Relationship between effect size and power As effect size increases power increases Sample size and power Increase in sample size decreases standard error thereby increases power Other ways to increase power Increase effect size sample size and alpha level Decrease beta population standard deviation and population standard error TABLE 83 A Summary of Factors That Increase Power the probability of rejecting a false null hypotheSIs To increase power Increase Decrease d Effect size 3 Type II error n Sample size 5 Standard deviation 0c Type I error GM Standard error Clicker Question 0 The power of a statistical test is the probability of o Rejecting a false null hypothesis Summary 0 Our rst statistical test 0 onesample ztest a does M differ from p o Directionality of hypothesis affects signi cance 0 Twotailed tests split margin of certainty between extremes of the distribution 0 Therefore more stringent 0 Two types of mquot in hypothesis testing 0 Type I error CL false negative incorrectly accept H0 0 Type II error B false positive incorrectly reject Ho 0 Statistical power ability to detect true effect 0 Correctly reject H0 0 power 1 3 0 Statistical power depends on o a increase a greater power 0 effect size increase a greater power 0 sample size increase a greater power Homework for Chapter 8 Factual problems 3 4 Concept and application problems 14 18 22 28 Problems in research 29 31 32 34
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