Psych 230 Psychological Measurement and Statisticswolfp/Psych230_files/chptr17.pdf · Statistical...

Psych 230

Psychological Measurement and Statistics

Pedro Wolf

December 9, 2009

This Time….

• Non-Parametric statistics

• Chi-Square test– One-way– Two-way

Statistical Testing

1. Decide which test to use2. State the hypotheses (H0 and H1)3. Calculate the obtained value - calculate r4. Calculate the critical value (size of α)5. Make our conclusion

1. Decide which test to use• Are we comparing a sample to a population?

– Yes: Z-test if we know the population standard deviation– Yes: One-sample T-test if we do not know the population std dev– No: Keep looking

• Are we looking for the difference between samples?– Yes: How many samples are we comparing?

• Two: Use the Two-sample T-test– Are the samples independent or related?

» Independent: Use Independent Samples T-test» Related: Use Related Samples T-test

– More than Two Groups: Keep looking• How many independent variables

– One: Use a One-way ANOVA– Two: Use a Two-factor ANOVA

– No: Keep looking• Are we looking for the relationship between variables?

– Yes: Use the Correlation test• Are we examining the frequencies of categorical, mutually exclusive

classes?– Yes: How many variables are we examining?

• One: Use one-way Χ2• Two: Use two-way Χ2

Assumptions• Mutually exclusive categories– One category per subject

• Independence of observations– equivalent criteria for entrance into

each category• Sufficient Sample Size– General rule: none of the expected

frequencies be less than five

Chi-Square

• Chi-Square is used when subjects are measured using a nominal variable– gender; political preference; handedness; nationality….

• With these variables, we do not measure an amount, but instead we count the frequency of observations in each of the categories– are psychology majors more likely to vote Republican or

Democrat?

• We are interested in proportions

Chi-Square

• In a Chi-Square test, we compare the actual frequencies of occurrence with what we would expect to happen by chance

• Two types of Chi-Square test• One-way: we have one variable• Two-way: we have two variables

One-Way Chi-Square

• We use this test when data consist of frequencies spread amongst different categories of a single variable– Are men or women more likely to smoke?– Is Coke or Pepsi more preferred by U.S. soda

drinkers?– Are different ethnic groups well represented

at the UofA?

Problem

• In a Psyc 230 class, are there equal numbers of people from rural, suburban and urban backgrounds?– Rural: 23– Suburban: 119– Urban: 50

• 3 levels, but still just one variable– One-way Chi-Square

2. State the Hypotheses

• H0 : all frequencies are equal– Rural, suburban and urban backgrounds are

equally represented in the class

• HA : not all frequencies are equal– Rural, suburban and urban backgrounds are

not equally represented in the class

3. Calculate the obtained value (χ2o b t )

Rural Suburban Urban TotalObserved (fo) 23 119 50 192

Expected (fe)



Expected (fe) 192/3 192/3 192/3

f e in eachcategory =Nk



Expected (fe) 64 64 64 192



Expected (fe) 64 64 64 192

χ 2 = Σ f o − f e 2f e



Expected (fe) 64 64 64 192

χ 2 = Σ f o − f e 2f e χ2o b t = Σ[(fo-fe)2 / fe ][(23-64)2 / 64] + [(119-64)2 / 64] + [(50-64)2 / 64] =

[(-41)2 / 64] + [(55)2 / 64] + [(-14)2 / 64] =

[ 1681 / 64] + [ 3025 / 64] + [ 196 / 64] =

26.266 + 47.266 + 3.062 = 76.594

4. Calculate the critical value

• Assume α=0.05• Always a two-tail test with Chi-square• Look up Table Q on page 572

– “Critical values of Chi-square: The χ2 tables”

• df = k - 1 = 3 - 1 = 2

• χ2c r i t for 2 degrees of freedom and α = 0.05 is 5.99

χ2 c r it and χ2 o b t

χ2 c r it = 5.99

χ2 o b t = 76.594

5. Make our Conclusion

• χ2c r it = 5.99• χ2o b t = 76.594

• χ2o b t is inside the rejection region, so we reject H0 and accept HA

• We conclude that there is a significant difference in frequencies between our groups– Rural, suburban and urban backgrounds are not

equally represented in our class

Problem

• In a Psyc 230 class, does the proportion of males and females in the class match the proportion of males and females who are psychology majors?– Psych majors: 75% female, 25% male– Females in the class: 142– Males in the class: 56


• The exact hypothesis will depend on the specific question we are asking

• H0 : frequencies in the class are equal to the psych population– 75% of our class are female, 25% are male

• HA : frequencies in the class are not equal to the psych population– 75% of our class are not female and 25% are

not male


Females Males TotalObserved (fo) 142 56 198

Expected (fe)



Expected (fe) (198*0.75) (198*0.25)



Expected (fe) 148.5 49.5 198



Expected (fe) 148.5 49.5 198

χ 2 = Σ f o − f e 2f e χ2o b t = Σ[(fo-fe)2 / fe ][(142-148.5)2 / 148.5] + [(56-49.5)2 / 49.5] =

[(-6.5)2 / 148.5] + [(6.5)2 / 49.5] =

[ 42.25 / 148.5] + [ 42.25 / 49.5] =

0.284 + 0.853 = 1.137


• Assume α=0.05• Always a two-tail test with Chi-square• Look up Table Q


• df = k - 1 = 2 - 1 = 1

• χ2crit for 1 degree of freedom and α = 0.05 is 3.84


χ2 c r it = 3.84

χ2 o b t = 1.137


• χ2c r it = 3.84• χ2o b t = 1.137

• χ2o b t is outside the rejection region, so we retain H0

• We conclude that there is no significant difference in frequencies between our groups as compared to the population of psychology majors– males and females are represented equally to their

distribution in the population

Problem - Your turn

• After conducting a survey of tastes with 30 subjects, researchers found the following preference for soda:– 18 people preferred Coke– 12 people preferred Pepsi

• Is there a significant difference in the number of people who preferred Coke over Pepsi? Use α=0.05.

χ 2 = Σ f o − f e 2f e f e in eachcategory= Nk


• H0 : all frequencies are equal– Coke and Pepsi are equally represented

• HA : not all frequencies are equal– Coke and Pepsi are equally not equally

represented


Coke Pepsi TotalObserved (fo) 18 12 30

Expected (fe) 15 15 30


χ 2 = Σ f o − f e 2f e χ2o b t = Σ[(fo-fe)2 / fe ][(18-15)2 / 15] + [(12-15)2 / 15] =

[(3)2 /15] + [(-3)2 /15] =

[ 9 / 15 ] + [ 9 / 15 ] =

0.6 + 0.6 = 1.2

Coke Pepsi TotalObserved (fo) 18 12 30

Expected (fe) 15 15 30


• Assume α=0.05• Always a two-tail test with Chi-square• Look up Table Q page 572


• df = k - 1 = 2 - 1 = 1

• χ2crit for 1 degree of freedom and α = 0.05 is 3.84


χ2 crit = 3.84

χ2 o b t = 1.2


• χ2c r it = 3.84• χ2o b t = 1.2

• χ2o b t is outside the rejection region, so we retain H0

• We conclude that there is no significant difference in frequencies between our groups– Coke and Pepsi are equally preferred

One-Way Chi-Square

• We use this test when data consist of frequencies spread amongst different categories of a single variable– Are men or women more likely to smoke?– Is Coke or Pepsi more preferred by U.S. soda

drinkers?– How different ethnic groups well represented

at the UofA?

Chi-Square

• What happens when we want to examine relationships between two variables?– Are higher or lower levels of self-esteem

more likely in athletes or non-athletes?– Is season of birth related to the likelihood of

suffering from Schizophrenia?– Do men and women have different

preferences for thin or heavy body-types?

Two-Way Chi-Square

• To answer questions concerning the relationship between two variables we use a two-way Chi-Square test

• Same logic as before - we will estimate what we would expect to see for each category if the null hypothesis was true, and then compare that to what we actually observed for each category

Problem

• Researchers are interested in the college experience of those who were the first in their family to attend college. One variable measured was if the students dropped out in their first semester. Is there a relationship between whether the student was the first in their family to go to college and whether they dropped out?

• Variables?

Problem

• First to go to college, dropped out: 15• First to go to college, didn’t drop out: 50• Not first to go to college, dropped out: 15• Not first to go to college, didn’t drop out:

120

Problem

• First to go to college, dropped out: 15• First to go to college, didn’t drop out: 50• Not first to go to college, dropped out: 15• Not first to go to college, didn’t drop out:

120First Not First Total

Dropped out 15 15 30Didn’t drop out 50 120 170

65 135 200


• H0 : there is no relationship between the two variables– whether you were first in your family to go to

college and whether you drop out are not related (the variables are independent)

• HA : there is a relationship between the two variables– whether you were first in your family to go to

college and whether you drop out are related (the relationship is not solely due to chance)


fo (fe) First Not First TotalDropped out 15 15 30

Didn’t drop out 50 120 170

65 135 200


f e =cell's row to ta l f o cell'sco lum nto ta l f o

N



65 135 200


f e =cell's row to ta l f o cell'sco lum nto ta l f o

N



65 135 200

Dropped out, first: fe = (30)(65) / 200 = 1950 / 200 = 9.75, round to 10

Dropped out, not first: fe = (30)(135) / 200 = 4050 / 200 = 20.25, round to 20

Didn’t drop out, first: fe = (170)(65) / 200 = 11050 / 200 = 55.25, round to 55

Didn’t drop out, not first: fe = (170)(135) / 200 = 22950 / 200 = 114.75, round to 115


fo (fe) First Not First TotalDropped out 15 (10) 15 (20) 30

Didn’t drop out 50 (55) 120 (115) 170

65 135 200


χ 2 = Σ f o − f e 2f e χ2o b t = Σ[(fo-fe)2 / fe ][(15-10)2 / 10] + [(15-20)2 / 20] + [(50-55)2 / 55] + [(120-115)2 / 115] =

[(5)2 / 10] + [(-5)2 / 20] + [(-5)2 / 55] + [(-5)2 / 115] =

[ 25 / 10] + [ 25 / 20] + [ 25 / 55] + [ 25 / 115] =

2.5 + 1.25 + 0.45 + 0.22 = 4.42

fo (fe) First Not First TotalDropped out 15 (10) 15 (20) 30

Didn’t drop out 50 (55) 120 (115) 170

65 135 200




• df = (number of rows - 1)(number of columns -1) = (2 - 1)(2 - 1) = 1

• χ2c r it for 1 degree of freedom and α = 0.05 is 3.84


χ2 c r it = 3.84

χ2 o b t = 4.42


• χ2c r i t = 3.84• χ2o b t = 4.42


• We conclude that there is a significant relationship between likelihood of dropping out of college and whether you were the first in your family to go– more likely to drop out if you were the first in your family

to attend college

Problem

• Researchers hypothesize that personality traits may be related to preference for color. In a study, researchers asked each subject to choose their favorite color from a choice of red, yellow, green and blue, and then tested whether the subject was either introverted or extroverted. Is there a relationship between these variables?

• Variables?

ProblemRed Yellow Green Blue

Introvert 10 3 15 22 Extrovert 90 17 25 18

ProblemRed Yellow Green Blue

Introvert 10 3 15 22 50 Extrovert 90 17 25 18 150

100 20 40 40


• H0 : there is no relationship between the two variables– your personality type (introvert or extrovert)

is not related to your favorite color (the variables are independent)

• HA : there is a relationship between the two variables– your personality type (introvert or extrovert)

is related to your favorite color (the variables are independent)

– the relationship is not solely due to chance


fo (fe) Red Yellow Green BlueIntrovert 10 3 15 22 50 Extrovert 90 17 25 18 150

100 20 40 40 200


f e =cell's row to ta l f o cell's co lum nto ta l f o

N


100 20 40 40 200


Introvert, Red : fe = (50)(100) / 200 = 5000 / 200 = 25

Introvert, Yellow : fe = (50)(20) / 200 = 1000 / 200 = 5

Introvert, Green : fe = (50)(40) / 200 = 2000 / 200 = 10

Introvert, Blue : fe = (50)(40) / 200 = 2000 / 200 = 10


100 20 40 40 200



N

Extrovert, Red : fe = (150)(100) / 200 = 15000 / 200 = 75

Extrovert, Yellow : fe = (150)(20) / 200 = 3000 / 200 = 15

Extrovert, Green : fe = (150)(40) / 200 = 6000 / 200 = 30

Extrovert, Green : fe = (150)(40) / 200 = 6000 / 200 = 30


100 20 40 40 200


fo (fe) Red Yellow Green BlueIntrovert 10 (25) 3 (5) 15 (10) 22 (10) 50 Extrovert 90 (75) 17 (15) 25 (30) 18 (30) 150

100 20 40 40 200


N


χ 2 = Σ f o − f e 2f e χ2o b t = Σ[(fo-fe)2 / fe ][(10-25)2 / 25] + [(3-5)2 / 5] + [(15-10)2 / 10] + [(22-10)2 / 10] + [(90-75)2 / 75] + [(17-15)2 / 15] + [(25-30)2 / 30] + [(18-30)2 / 30] =


100 20 40 40 200


χ 2 = Σ f o − f e 2f e χ2o b t = Σ[(fo-fe)2 / fe ][(-15)2 / 25] + [(-2)2 / 5] + [(5)2 / 10] + [(12)2 / 10] + [(15)2 / 75] + [(2)2 / 15] + [(-5)2 / 30] + [(-12)2 / 30] =

[225 / 25] + [4 / 5] + [25 / 10] + [144 / 10] + [225 / 75] + [4 / 15] + [25 / 30] + [144 / 30] =

9 + 0.8 + 2.5 + 14.4 + 3 + 0.27 + 0.83 + 4.8 = 35.6


100 20 40 40 200




• df = (number of rows - 1)(number of columns -1) = (2 - 1)(4 - 1) = 3

• χ2c r it for 3 degrees of freedom and α = 0.05 is 7.81


χ2 c r it = 7.81

χ2 o b t = 35.6


• χ2c r it = 7.81• χ2o b t = 35.6


• We conclude that there is a significant relationship between color choice and extroverted/introverted personality types

Final Exam

• Questions–Multiple-choice (30 questions @ 2 points each):

60 total– Short-answer (4 questions @ 10 points each): 40

total• Some questions from throughout the

semester• You’ll be provided with all required formulas

and tables• Remember to bring calculators and pencils

Populations and Samples

• A population is all possible members of the group of interest

• A sample is a subset of the population

Descriptive and Inferential Statistics

• Descriptive statistics– procedures which organize and summarize

sample data

• Inferential statistics– procedures for drawing inferences about

populations

Statistics and Parameters

• Statistic– a number that describes an aspect of a sample

of scores

• Parameter– a number that describes an aspect of a

population of scores• often inferred through sampling

Experimental Studies

• In a true experiment, the researcher actively changes or manipulates one variable and then measures participants’ scores on another variable to see if a relationship is produced– example: the effect of alcohol on stats test scores

• Two types of variable:– independent variable

• manipulated a variable the experimenter actually manipulates (e.g. treatment condition)

• subject a measurable aspect of the individual participants which the experimenter does not change (e.g. sex)

– dependent variable

Correlational Studies

• The researcher measures participants’ scores on two variables and then determines whether a relationship is present

Which Scale?

1. Does the variable have an intrinsic value?NO ==> Nominal

2. Does the variable have equal values between scores?NO ==> Ordinal

3. Does the variable have a real zero point?NO ==> IntervalYES ==> Ratio

Things you need to know• Be able to describe a normal distribution• Know the difference between a null and

alternative hypothesis• Know the difference between type 1 and type

two error– A Type I error is defined as rejecting H0 when H0 is actually true– A Type II error is defined as retaining H0 when H0 is false (and H1 is actually true)

• Correlation is not causation

Things you need to know(continued)

• A linear relationship forms a pattern on a scatterplot that fits a straight line

• In a positive linear relationship, as the scores on the X variable increase, the scores on the Y variable also tend to increase

• In a negative linear relationship, as the scores on the X variable increase, the scores on the Y variable tend to decrease

Things you need to know(continued)

• The assumptions of chi square–Mutually exclusive categories• One category per subject

– Independence of observations• equivalent criteria for entrance into each

category– Sufficient Sample Size• General rule: none of the expected frequencies

be less than five

Know how to

• use this decision tree • Are we comparing a sample to a population?

– Yes: Z-test if we know the population standard deviation– Yes: One-sample T-test if we do not know the population std dev– No: Keep looking

• Are we looking for the difference between samples?– Yes: How many samples are we comparing?

• Two: Use the Two-sample T-test– Are the samples independent or related?

» Independent: Use Independent Samples T-test» Related: Use Related Samples T-test

– More than Two Groups: Keep looking• How many independent variables

– One: Use a One-way ANOVA– Two: Use a Two-factor ANOVA

– No: Keep looking• Are we looking for the relationship between variables?

– Yes: Use the Correlation test

• Are we examining the frequencies of categorical, mutually exclusive classes?– Yes: How many variables are we examining?

• One: Use one-way Χ2

• Two: Use two-way Χ2

Know how to (continued)

• calculate an arithmetic mean and standard deviation

• use the various tables in the back of the book related to material covered .

• create a scatter plot correctly from raw data

Know how to (continued)

• use the formula to calculate r.• calculate r2 • calculate Y’ or X’• draw a regression line when given

only the slope and intercept• Calculate the obtained value (χ2o b t)– both one-way and two way

Know How to interpret ANOVA results and terms

• If I give you a complete F table tell me what that means conceptually

• If I give you the results of a post-hoc test interpret those results

Psych 230 Psychological Measurement and StatisticsSlide 2This Time….Statistical Testing1. Decide which test to useAssumptionsChi-SquareSlide 8One-Way Chi-SquareProblem2. State the Hypotheses3. Calculate the obtained value (2obt)Slide 13Slide 14Slide 15Slide 164. Calculate the critical value2 crit and 2 obt5. Make our ConclusionSlide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Problem - Your turnSlide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Two-Way Chi-SquareSlide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Populations and SamplesDescriptive and Inferential StatisticsStatistics and ParametersExperimental StudiesCorrelational StudiesWhich Scale?Things you need to knowThings you need to know (continued)Slide 74Know how to Know how to (continued)Slide 77Slide 78

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