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# K 300 PSY

IU

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This 0 page Class Notes was uploaded by Kole Corwin on Sunday November 1, 2015. The Class Notes belongs to PSY at Indiana University taught by Luiz Pessoa in Fall. Since its upload, it has received 9 views. For similar materials see /class/233465/psy-indiana-university in Psychlogy at Indiana University.

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Date Created: 11/01/15

Probability allows us to establish a formal M between populations and samples Helps answer the following question Which samples can be likely obtained given a certain population Inferentia statistics utilizes this information to make inferences about populations from samples What is Probability Relationship between samples and populations lt INFERENTIAL STATISTICS Population Sample PROBABILITY What kinds of samples are likely Defining Probability For a situation in which several different outcomes are possible the probability of any speci c outcome is de ned as a fraction or a proportion of all the possible outcomes Probability proportion of outcome Given an outcome A probability of A Number of outcomes classed as A Total number of outcomes Examples coin toss deck of cards Defining Probability What is the probability of selecting a king from a deck of cards Can be restated as What proportion of the whole deck consists of kings In both cases 4 out of 52 pking 452 Probabilities can be expressed as fractions decimals or as percentaqes 452 0769 769 Probability Range of Values Probabilities range from 0 to 1 Probabilities must have Qositive values or zero Probability and Random Sampling For a random sample these two conditions must be met 1 Each individual has an equal chance of being selected 2 If more than one individual is selected probabilities stav constant from one selection to the next Consider coin tosses each time pheads ptais 05 even if you have observed heads 10 times in a row Equal Chance 1 Each individual has an equal chance of being selected For a population with N individuals each individual must have the same probability p1N Constant Probability 2 If more than one individual is selected there must be constant Qrobabiity for each selection Consider the selection of n 2 cards p jack of diamonds 152 Second draw assuming you are holding the rst card What is the probability of obtaining the jack of diamonds p jack of diamonds 151 if rst card was not jack of diamonds p jack of diamonds 0 if it was Constant probability requires gmplinq with re cement Probability and Frequency Distribution Graphs Probabilities correspond to proportions of scores as shown in frequency distribution graphs Example For the population of scores shown below what is the probability in a random draw of obtaining a score greater than 4 pXgt4 210 pXlt5 810 3 Frequency m The Normal Distribution Diagram The Normal Distribution Properties 1 Symmetrical and unimoda mean median mode 2 Mean median divides distribution in half 3 Most of the scores are found around the mean Extreme scores are found in the two tails The Normal Distribution Proportions of areas within the normal distribution can be quanti ed using zscores 3413 1359 228 The Normal Distribution The normal distribution is symmetrical This means that the proportions on both sides of the mean are identical A normal distributions have the same proportions The Normal Distribution Body height has a normal distribution with u 68 inches and 039 6 If we select one person at random what is the probability of selecting a person taller than 80 pXgt80 Xu 8068 2 T T 12620 pzgt20 pX gt 80 pz gt 20 228 The Normal Distribution A graphical representation of the same problem b l I I I I X I I I X 68 74 80 68 74 80 M M Z 0 200 The Unit Normal Table Given the standard proportions of normal distributions we can give probabilities for zscores with whole number values But what about fractional zscores Need to use the unit normal table The Unit Normal Table How the table is organized A E C gt D Z Propomon Pmpm on Pmpm39 on in Bmdv in 10 Beiween M9011 and z 000 5000 5000 0000 001 5040 4960 0040 002 5080 4920 0080 003 5120 0120 W W W 5832 4168 W 022 5871 4129 0871 023 5910 4090 0910 024 5948 4052 0948 025 5987 4013 0987 026 6026 3974 1026 027 6064 3936 1064 028 6103 3897 1103 029 6141 3859 1141 030 6179 3821 1179 031 6217 3783 1217 032 6255 3745 1255 033 6293 3707 1293 034 6331 3669 1331 A Mean 2 A Mean 2 A Mean 2 The Unit Normal Table Things to remember when using the unit normal table 1 Symmetrical only positive 2 scores are tabulated 2 Proportions are always positive 3 Section gt 50 larger body 4 Section lt 50 smaller tail 5 Body tail 100 100 In a graph area greater than area to the right of area smaller than area to the left of The Unit Normal Table How do we use the unit normal table Example Find the proportion of the normal distribution corresponding to a z score greater than z100 Draw a sketch 01 Then look up the value in the table The Unit Normal Table For a normal distribution what is the probability of selecting a z score value greater than z10 0 The Unit Normal Table For a normal distribution what is the probability of selecting a z score value less than z15 b O 150 The Unit Normal Table What proportion of the normal distribution corresponds to the tail beyond negative 2 05 C The Unit Normal Table What z score separates the top 10 from the remainder of the distribution O m Wm m End 4n ywm h 5mm Mm w The Unit Normal Table What zscore values form the boundaries that separate the middle 60 of the distribution from the rest of the scores From Specific Scores to The Unit Normal Table You are asked a probability associated with a speci c X value as opposed to a zscore For a normal distribution with u 100 and 039 15 give the probability of selecting an individual whose score is less than 130 proportion of individuals with a score less than 130 From Specific Scores to The Unit Normal Table Follow this procedure 1 Make a rough sketch 1 and 039 2 Locate and mark speci c score X 3 Shade appropriate proportion 4 Transform X value into zscore 5 Look up value for proportion in unit normal table using zscore From Specific Scores to The Unit Normal Table p X lt 130 X39 0 130 100 15 3015 20 2 pX lt 130 pz lt 20 9772 9772 19 M Cy In w I5 4 13 Weltclan t par w mum mprrtiur Pazpu ni m 39 in Bus 1 T2417 F hum l 1 Err y m i 5 39 en Mam f F1931 Uc m 1 539 3041 I77 39L j 4 J 7 4 Huif 35 3m 773 117 I VE i 539 157 06 33 quotquot7 01 J 1733 l 9371 Hhim n n 1m 0733 u 4715 LIFO QR12 4383 2 DJ 3911 WVquot 47 Finding Probabilities between Two Scores Average speed is 58 mph and standard deviation is 10 Assuming that the distribution is normal what proportion of the cars are traveling between 55 and 65 mph A 55 65 958 I I 30 O 70 Finding Probabilities between Two Scores Average speed is 58 mph and standard deviation is 10 Assuming that the distribution is normal what proportion of the cars are traveling between 55 and 65 mph p55ltXlt65 Xu 5558 2 T T 310 3 Xu 6558 ZT 1 0 7107 p 03 lt z lt 07 11792580 3759 1 tail 03 tail 07 The Unit Normal Table w av my my 0 an hmmmm i mwmnn mm mm pmpmm lquot M m39rm Ezlurzn um and I m m Helwrm Mm m mm m 1 0937 mm mm mm mm 393K mas mm 3397 nu mm 3391 x m owv 1x2 m 0299 1783 an 0279 74 ms 0 w I 37117 1293 msy n 54 om saw ml um quot15 was 3532 USE msu n 36 st 35 nor um Im A 2557 Im 0517 m mm 3520 MN 41557 n 39 517 M lt17 05 mo 05 am my 0636 n 4 nsm 3119 mm mm 042 my 1172 um um 43 mm mr mm 075 044 mm mm Hun mu v45 am we m u a 577 mu m Wm m End 4n ywm h 5mm Mm w Transform a tians X z score formula Z sCOre Unlr normal To ble Proponions 0r Probabilities Another example SAT scores form a normal distribution with u 500 and a 100 What is the minimum score to be in the top 15 of the SAT distribution Draw a sketch Top 15 Another example SAT scores form a normal distribution with u 500 and 0 100 What is the minimum score to be in the top 15 of the SAT distribution Draw a sketch Look at table Which column 2 104 Now need to get X based on 2 X u 20 500 104100 500 104 604 Binomial Distribution Let s skip section 64 on the binomial distribution 1 Goals of the Course 2 The What and Why of Statistics 3 The Syllabus The Course Syllabus Available on Oncourse Syllabus is approximate only Note syllabus may be subject to change including exam dates Changes will be announced in class and via email Goals of the Course 1 Get a good grasp of the basic concepts and strategies of statistics 2 Be able to apply statistical methods in their appropriate context 3 Become a better critic and skeptic of statistics A Word About Phobias Let s face it Most of you aren t here because you thought statistics would be a fun elective How to succeed 1 For the extreme math phobics Come see me and TA regulary 2 Keep up with the material 3 Test yourself Work probems yourself after we39ve done them in class A Word About Phobias 1 All statistical procedures were developed to serve a purpose 2 If you understand why a new procedure is needed you will find it much easier to learn the procedure After each classchapter try to read the Preview section at the beginning of each book chapter Why Sta tistiCS 1 Statistics is all around us 2 Science is based on observation statistics allows us to organize summarize and interpret empirical data Examples 1 In a psychological experiment we need to determine a human subject s reaction time The measurements we obtain vary a great deal from one trial to the next What can we do to get a reliable estimate 2 A drug company has developed a new substance that they claim reduces blood pressure How do we test this claim 3 The mayor of a large city must decide whether to build an extension of the downtown highway system or not The mayor is concerned about voter support How does the mayor nd out what people think Populations and Samples Observations are usually made on individuals A population is the set of all the individuals of interest Examples All students that apply to IU a given year All students that are accepted at IU a given year A study always needs to specify the target population Why Conclusions only apply to that population Populations and Samples A sample is a set of individuals selected from a population usualy samples should be representative not biased Although the sample needs to come from the target population sample size can vary Why For instance 1 because I may only have resources to collect data from 100 or 1000 individuals 2 Theoretical results may indicate that data from 100 individuals are suf cient for adequate results Fundamental Logic of Statistical Reasoning THE POPULATION All of The individuals of inTeresT in feren 7 The resulTs from The sample are generalized To The populaTion THE SAMPLE The individuals selecTed To par TicipaTe in The research sTudy sampling The sample is selecTed from The populaTion Populations and Samples Population of individuals gt population of scores Sample of individuals gt sample of scores Observation measurement datum score raw score A parameter describes a population A statistic describes a sample For instance the mean of the population And the mean of the sample Descriptive Statistics Descrigtive statistics are used to summarize organize and simplify data some examples Descriptive Statistics Tornadoes Descriptive Statistics Tornadoes 15 15 V Average Number 125 of Tanadoa Per Mmlh l u A 5 A i 199 Oklahoma Climatological Survey All rights reserved Descriptive Statistics Tornado es m Average Number 39 of Tmnadees Per Hour of me Day an 199 Oklahoma Climexelegiee Survey All rights reserved Inferential Statistics Inferential statistics study samples and allow generalizations inferences about the population from which the sample was obtained assuming the sample was representative For example I want to use the data from 100 students to make conclusions about all of the incoming students ofIU By the end of the course you should have an understanding of why this works and its limitations Inferential Statistics Sampling error is the discrepancy between a sample statistic and the corresponding DODuation parameter Keep the sampling error small Use large samples Use random sampling Fundamental Logic of Statistical Reasoning THE POPULATION All of The individuals of inTeresT in feren 7 The resulTs from The sample are generalized To The populaTion THE SAMPLE The individuals selecTed To par TicipaTe in The research sTudy sampling The sample is selecTed from The populaTion Example of Sampling Error Papu1a1ion af10110 ao11ege srvdents P00u1a11arr Parame1ers Average Age 213 years Average 19 1125 65 Female 35 We Greater sampling error for IQ Samp1ei1 Samp1ei9 Eric 10m Jessica Kr rsien Laura Sara Karen Andrew Br1an Jahrr Sample a aiu Sarng1eS1a1r srics Average Age 1916 Average Age 2014 Average 19 10416 Average 19 1142 60Fema1e40Ma1e 411 Female60Mab Inferential Statistics Sampling error is the discrepancy between a sample statistic and the corresponding DODuation parameter Keep the sampling error small Use large samples Use random sampling 20 Fundamental Logic of Statistical Reasoning Let s try an example THE POPULATION All of The individuals of inTeresT inferencf Rimming The resulTs Thesqmple is selecied from To The populaTion The popu39onon THE SAMPLE The individuais selecTed To pariicipoie in The research sTudy 21 height 3 Population K300 Mean 669 40 60 tudent a uv Sample 1 7 ean 68565 75 100 22 Statistics in the Context of Research as opposed to desc stats Step 1 Experiment Compo39e mo Teachhg mehhocs Dom Tamas home s39ddeh s heam some e am IeA Samp eB S p Taught by Method A Taught by Melhod B 75 72 79 68 70 73 77 77 75 77 67 72 70 77 75 70 78 75 68 70 77 74 7O 78 72 74 69 72 77 74 87 7O 73 70 70 77 683816 5h SampreA SampreB Devcrr39ph39ve ha s cs39 Organize ahasrrnplir v I I I 70 80 85 65 75 80 35 Average Average score76 score71 Steps Thesarhpledarashowaepohtditrerenoe Inferenrral between he rwo reaching methodsr However harm fherearerwoways w eresuhs Interprer resuris rr There acTuahr no drherence erweeh andrhesampre differehr ancesamprngerrorr 2r Iherehveeh rhehva Lr esampledara accurarelv re ecrrhrs driererrcer CS inexperim mal research The goal at hferehh ar star rsrics hi0 harp researchers decide berween the Mo inrerprera ore Statements are probabilistic Correlational Method Correlational method Two variables are observed and Checked to see if a relationship exists 7 8 9 10 WokeupTime H 25 Correlational Method Correlational method Two variables are observed and checked to see if a relationship exists Does eary wakeup time cause better academic performance 9 Correlation does not imply causation 26 Experimental Method Experimental method Goal is to establish causal relationships between variables Requires manipulation and control conditions The independent variable is the one that is manipulated The dependent variable is the one that is observed 27 Experimental Method Variable 1 Room iemperaiure ihe independeni variable Manipulaied lo cieaie iwo 70 Room 90 Room ireaimeni condiiions 17 12 19 10 16 14 12 15 Variable 2 Memory scores 17 i3 ihe dependeni variable 18 12 Measured in each oiihe 15 11 ireaimeni condiiions 16 13 Any difference Experimental Method Control condition receives no treatment or placebo Experimental condition receives treatment Confoundinq variable uncontrolled a source of error in interpretation Example Back to the room temperature experiment 29 Discrete and Continuous Variables A discrete variable consists of separate indivisibe categories e 9 number of male children in family A continuous variable is divisible into an in nite number of fractional parts eg weight of male children in family 30 Scales of Measurement When collecting data we need to make measurements How do we measure things By putting them into categories Qualitative By using numbers Quantitative Different kinds of scales nominal ordinal interval ratio 31 Scales of Measurement 0 Nominal Set of categories that have different names more than or less than not defined Major Math Stats Physicis o Ordinal Organized in an ordered sequence You can determine the direction of difference ie order Groups lower middle upper socioeconomic class 32 Scales of Measurement 0 Interval Ordered categories that are all intervals of exactly the same size Differences are well defined but ratios of magnitudes are not meaningful Fahrenheit Celsius o What is missing from Fahrenheit 0 A true ie not arbitrary zero point 0 Twice the temperature does not mean twice the molecular energy 0 Temperature Kelvin thermodynamic temperature minimal molecular movementquot 33 Scales of Measurement 0 Ratio Interval scale with the additional feature of an absolute zero point Ratios are well defined Height age reaction time temperature on the Kelvin scale 34 Example a Buildings of Interest 39 A B C D 204 ft 104 ft 180 ft lOZ ft b Ordinal Measure 1 39 2 3 j 4 D B C A c Interval B is 2 ft taller than D Measure J 2 ft 39 L 104 h Crlterion 7 7 100 f1 D d Ratio Measure A is twice as tall as D 180 f1 204 ft llmlllw 8 From Jaccard and Becker 5th ed Fig 11 35 Statistical Notation One score X Two scores X Y Number of scores sample n Number of scores population N Summation 2X sum of X think of summing over a column in a spreadsheet Example X3174 ZX317415 2X291491675 Note 2X 5 2X 5 2X2 2X2 36 Statistical Notation Note 2X 5 2X2 ZX5 202 X 3 1 7 4 2X 15 2X5 2 4 2 1 5 2X5 ZX5 15 5 10 2X2 2X x 2X 15x 15 225 2X2 75 37

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