t-Static t-test is used to test hypothesis about an unknown population mean, µ, when the value of...

t-Statict-Static t-testt-test is used to test is used to test hypothesis about anhypothesis about an unknown population unknown population mean, µ, mean, µ, when the when the value of value of σσ or or σσ²² is is unknownunknown. . 22

Degrees of FreedomDegrees of Freedomdf=n-1df=n-1

33

Assumption of the Assumption of the t-test t-test (Parametric Tests)(Parametric Tests)

1.The Values in the sample must 1.The Values in the sample must consist ofconsist of independent independent observations observations

2. The population sample must be 2. The population sample must be normal normal

3. use a 3. use a large sample large sample n ≥ 30n ≥ 3044

Inferential StatisticsInferential Statistics t-Statistics:t-Statistics: There are different types of t- StatisticThere are different types of t- Statistic 1. 1. SingleSingle (one) Sample t-statistic (test) (one) Sample t-statistic (test) 2. 2. TwoTwo independent sample t-test, independent sample t-test,

Matched-Subject ExperimentMatched-Subject Experiment, or , or Between Subject DesignBetween Subject Design

3.3.RepeatedRepeated MeasureMeasure Experiment, or Experiment, or Related/Paired Related/Paired SampleSample t-test t-test

55

Hypothesis Testing Hypothesis Testing Step 3: Computations/ Calculations or Collect Step 3: Computations/ Calculations or Collect

Data and Compute Sample StatisticsData and Compute Sample StatisticsFYI FYI Z Score for ResearchZ Score for Research

66

Hypothesis TestingHypothesis TestingStep 3: Computations/ Calculations or Collect Step 3: Computations/ Calculations or Collect

Data and Compute Sample StatisticsData and Compute Sample Statistics

77

Calculations for Calculations for t-testt-test

M-M-μμ t= st= s

SSmm S Sm= orm= or

SSmm== M-M-μμ √n √n

t t

M=t.SM=t.Sm+m+μμ

μμ=M- S=M- Sm.tm.t

88

2 /S n

FYI FYI Variability VariabilitySS,SS, Standard Deviations and VariancesStandard Deviations and Variances

X X σ² = ss/N σ² = ss/N PopPop 1 1 σ = √σ = √ss/Nss/N 22 4 s = √ss/df 4 s = √ss/df 5 5 s² = ss/n-1 or ss/df s² = ss/n-1 or ss/df Sample Sample

SS=SS=ΣxΣx²-(Σx)²/N²-(Σx)²/N

SS=SS=ΣΣ(( x-x-μμ))²²

Sum Sum of of SquaredSquared DeviationDeviation from from MeanMean99

d=d=Effect SizeEffect SizeUse Use SS instead of instead of σσ for for t-testt-test

1010

Cohn’s d=Cohn’s d=Effect SizeEffect SizeUse Use SS instead of instead of σσ for for t-testt-test

1111

d = (M - µ)/sd = (M - µ)/s

s= (M - µ)/ds= (M - µ)/d

M= (d . s) + µM= (d . s) + µ

µ= (M – d) sµ= (M – d) s

Percentage of Variance Accounted Percentage of Variance Accounted for by the Treatment (similar to for by the Treatment (similar to Cohen’s d) Also known as Cohen’s d) Also known as ωω² ²

Omega SquaredOmega Squared

1212

22

2

tr

t df

percentage of Variance accounted percentage of Variance accounted for by the Treatment for by the Treatment

Percentage of Variance Explained Percentage of Variance Explained r²=0.01--------r²=0.01-------- Small Effect Small Effect

r²=0.09--------r²=0.09-------- Medium Effect Medium Effect

r²=0.25--------r²=0.25-------- Large Effect Large Effect

1313

2r2r

ProblemsProblems Infants, even Infants, even newbornsnewborns prefer to look at prefer to look at attractive faces (Slater, et al., 1998). In the attractive faces (Slater, et al., 1998). In the study, infants from study, infants from 1 to 6 days old 1 to 6 days old were shown were shown two photographs two photographs of women’s face. Previously, a of women’s face. Previously, a group of adults had rated one of the faces as group of adults had rated one of the faces as significantly more attractive than the other. The significantly more attractive than the other. The babies were positioned in front of a screen on babies were positioned in front of a screen on which the photographs were presented. The pair which the photographs were presented. The pair of faces remained on the screen until the baby of faces remained on the screen until the baby accumulated a total of accumulated a total of 20 seconds 20 seconds of looking at of looking at one or the other. The number of seconds looking one or the other. The number of seconds looking at the attractive face was recorded for each at the attractive face was recorded for each infant. infant. 1414

ProblemsProblems Suppose that the study used a sample of Suppose that the study used a sample of n=9n=9

infants and the data produced an average of infants and the data produced an average of M=13M=13 seconds for attractive face with seconds for attractive face with SS=72.SS=72.

Set the level of significance at Set the level of significance at αα=.05=.05

for for two tails two tails Note that all the available information comes Note that all the available information comes

from the sample. Specifically, from the sample. Specifically, we do not know we do not know the population mean the population mean μμ or the population or the population standard deviation standard deviation σσ. .

On the basis of this sample, can we On the basis of this sample, can we conclude that infants prefer to look at conclude that infants prefer to look at attractive faces?attractive faces? 1515

Null HypothesisNull Hypothesis t-Statistic:t-Statistic: If the Population mean or If the Population mean or µ and the µ and the

sigma are unknown sigma are unknown the statistic of the statistic of choice will be choice will be t-Statict-Static

1.1. Single Single (one) Sample t-statistic (test)(one) Sample t-statistic (test) Step 1Step 1 HH00 : µ : µ attractive attractive = 10 seconds = 10 seconds HH11 : µ : µ attractiveattractive ≠ 10 seconds≠ 10 seconds

1616

ProblemsProblems A psychologist has prepared an “Optimism A psychologist has prepared an “Optimism

Test” that is administered yearly to Test” that is administered yearly to graduating college seniors. The test graduating college seniors. The test measures how each graduating class measures how each graduating class feels about its future. The higher the score feels about its future. The higher the score , the more optimistic the class. Last year’s , the more optimistic the class. Last year’s class had a mean score of class had a mean score of μμ=15=15. A . A sample of sample of n=9 n=9 seniors from this year’s seniors from this year’s class was selected and tested.. class was selected and tested..

1717

ProblemsProblems The scores for these seniors are The scores for these seniors are 7, 12, 11, 7, 12, 11,

15, 7, 8, 15, 9, and 615, 7, 8, 15, 9, and 6, which produced a , which produced a sample mean of sample mean of M=10M=10 with with SS=94. SS=94.

On the basis of this sample, can the On the basis of this sample, can the psychologist conclude that this year’s psychologist conclude that this year’s class has a class has a differentdifferent level of optimism?level of optimism?

Note that this hypothesis test will use a t-Note that this hypothesis test will use a t-statistic because the statistic because the population variance population variance σσ² is not known. ² is not known. USE SPSSUSE SPSS

Set the level of significance at Set the level of significance at αα=.01 for two tails =.01 for two tails

1818

Null HypothesisNull Hypothesis t-Statistic:t-Statistic: If the Population mean or If the Population mean or µ and the µ and the

sigma are unknown sigma are unknown the statistic of the statistic of choice will be choice will be t-Statict-Static

1.1. Single Single (one) Sample t-statistic (test)(one) Sample t-statistic (test) Step 1Step 1 HH00 : µ : µ optimism optimism = 15 = 15 HH11 : µ : µ optimismoptimism ≠ 15≠ 15

1919

Chapter 10Chapter 10TwoTwo Independent Sample t-testIndependent Sample t-test Matched-Subject ExperimentMatched-Subject Experiment, or , or

Between Subject DesignBetween Subject DesignAn independent-measures studyAn independent-measures study

uses a separate sample to uses a separate sample to represent each of the represent each of the populations or treatment populations or treatment conditions being compared.conditions being compared.

2020

TwoTwo Independent Sample t-testIndependent Sample t-testNull HypothesisNull Hypothesis:: If the Population mean or µ is unknown If the Population mean or µ is unknown

the statistic of choice will be the statistic of choice will be t-Statict-Static Two Two independent sample t-test, independent sample t-test, Matched-Matched-

Subject ExperimentSubject Experiment, or Between Subject , or Between Subject DesignDesign

HH00 : µ : µ1 1 -µ-µ2 2 = 0= 0 HH11 : µ : µ11 -µ -µ22 ≠ 0 ≠ 0

2121

ProblemsProblems Research results suggest aResearch results suggest a relationship relationship

Between the TV viewing habits of 5-year-old children and their future performance in high school. For example, Anderson, Huston, Wright & Collins (1998) report that high school students who regularly watched Sesame Street as children had better grades in high school than their peers who did not watch Sesame Street.

2222

ProblemsProblems The researcher intends to examine The researcher intends to examine

this phenomenon using a sample of this phenomenon using a sample of 20 high school students. She first 20 high school students. She first surveys the students’ s parents to surveys the students’ s parents to obtain information on the family’s TV obtain information on the family’s TV viewing habits during the time that the viewing habits during the time that the students were 5 years old. Based on students were 5 years old. Based on the survey results, the researcher the survey results, the researcher selects a sample of selects a sample of n=10n=10

2323

ProblemsProblems students with a history of watching students with a history of watching

“Sesame Street“ and a sample of “Sesame Street“ and a sample of n=10 n=10 students who did not watch the students who did not watch the program. The program. The average high school average high school grade grade is recorded for each student is recorded for each student and the data are as follows: and the data are as follows: Set the Set the level of significance at level of significance at αα=.05=.05

for two tails for two tails

2424

ProblemsProblemsAverage High School GradeAverage High School Grade

Watched Sesame St (1). Watched Sesame St (1). Did not Watch Sesame St.(2)Did not Watch Sesame St.(2)

86 86 90 90

87 87 89 89

9191 82 82

9797 83 83

98 98 85 85

99 99 7979

9797 83 83

94 94 86 86

8989 81 81

9292 92 92

nn11=10 =10 n n22=10=10

MM11=93 =93 M M22= 85= 85

SSSS11=200 =200 SS SS22=160=1602525

TwoTwo Independent Sample t-testIndependent Sample t-testNull HypothesisNull Hypothesis:: Two Two independent sample t-test, independent sample t-test, Matched-Matched-

Subject ExperimentSubject Experiment, or Between Subject , or Between Subject Design-Design- non-directional or two-tailed non-directional or two-tailed testtest

Step 1.Step 1. HH00 : µ : µ1 1 -µ-µ2 2 = 0= 0 HH11 : µ : µ11 -µ -µ22 ≠ 0 ≠ 0

2626

TwoTwo Independent Sample t-testIndependent Sample t-testNull HypothesisNull Hypothesis:: Two Two independent sample t-test, independent sample t-test, Matched-Matched-

Subject ExperimentSubject Experiment, or Between Subject , or Between Subject Design -Design - directional or one-tailed testdirectional or one-tailed test

Step 1.Step 1. HH00 : µ : µ Sesame St . Sesame St . ≤ µ ≤ µ No Sesame St. No Sesame St.

HH11 : µ : µ Sesame St. Sesame St. >> µ µ No Sesame St.No Sesame St.

2727

Measuring d=Measuring d=Effect Size for the Effect Size for the independent measuresindependent measures

2828

2

1 2M Md

S p

Percentage of Variance Accounted Percentage of Variance Accounted for by the Treatment for by the Treatment (similar to (similar to

Cohen’s d) Also known as Cohen’s d) Also known as ωω² Omega ² Omega Squared Squared and Coefficient of and Coefficient of

DeterminationDetermination

2929

22

2

tr

t df

We use the We use the Point-Biserial Correlation Point-Biserial Correlation when one of our variable is dichotomous, in when one of our variable is dichotomous, in

this case (1) watched Sesame St. this case (1) watched Sesame St. (2) and didn’t watch Sesame St. (2) and didn’t watch Sesame St.

3030

22

2

tr

t df

ProblemsProblems In recent years, psychologists have In recent years, psychologists have

demonstrated repeatedly that using mental demonstrated repeatedly that using mental images can greatly improve memory. Here we images can greatly improve memory. Here we present a hypothetical experiment designed to present a hypothetical experiment designed to examine this phenomenon.examine this phenomenon.

The psychologist first prepares a list of The psychologist first prepares a list of 40 pairs 40 pairs of of nounsnouns (for example, dog/bicycle, grass/door, (for example, dog/bicycle, grass/door, lamp/piano). Nextlamp/piano). Next, two groups of participants , two groups of participants are obtained (two separate samples). are obtained (two separate samples). Participants in one group are given the list for 5 Participants in one group are given the list for 5 minutes and instructed to memorize the 40 minutes and instructed to memorize the 40 noun pairs. noun pairs. 3131

ProblemsProblems Participants in another group receive the same Participants in another group receive the same

list of words, but in addition to the regular list of words, but in addition to the regular instruction, they are told to form a instruction, they are told to form a mental image mental image for each pair of nouns for each pair of nouns (imagine a dog riding a (imagine a dog riding a bicycle, for example). Later each group is given bicycle, for example). Later each group is given a memory test in which they are given the first a memory test in which they are given the first word from each pair and asked to recall the word from each pair and asked to recall the second word. The psychologist records the second word. The psychologist records the number of words correctly recalled for each number of words correctly recalled for each individual. The data from this experiment are as individual. The data from this experiment are as follows: follows: Set the level of significance at Set the level of significance at αα=.01=.01

for two tails for two tails 3232

ProblemsProblemsData (Number of words recalled)Data (Number of words recalled)

Group 1 (Images) Group 1 (Images) Group 2 (No Images) Group 2 (No Images)

19 19 23 23

20 20 22 22

2424 15 15

3030 16 16

31 31 18 18

32 32 1212

3030 16 16

27 27 19 19

2222 14 14

2525 25 25

nn11=10 =10 n n22=10=10

MM11=26 =26 M M22= 18= 18

SSSS11=200 =200 SS SS22=160=1603333

3.3.RepeatedRepeated Measure Experiment, Measure Experiment, or or Related/PairedRelated/Paired Sample t-test Sample t-test Within Subject Within Subject Experiment DesignExperiment Design

A A single sample single sample of individuals of individuals is measured is measured more than once more than once on the same dependent on the same dependent variable. The variable. The same subjects same subjects are used in all of the treatment are used in all of the treatment conditions.conditions.

3434

Null HypothesisNull Hypothesis t-Statistics:t-Statistics: If the Population mean or µ is unknown If the Population mean or µ is unknown

the statistic of choice will be the statistic of choice will be t-Statistict-Statistic 3.Repeated Measure Experiment, or 3.Repeated Measure Experiment, or

Related/Paired Sample t-testRelated/Paired Sample t-test For For Non-directional or two tailed testNon-directional or two tailed test Step. 1Step. 1 HH00 : µ : µDD = 0= 0 HH11 : µ : µD D ≠ 0≠ 0

3535

Null HypothesisNull Hypothesis t-Statistics:t-Statistics: If the Population mean or µ is unknown If the Population mean or µ is unknown

the statistic of choice will be the statistic of choice will be t-Statistict-Statistic 3.Repeated Measure Experiment, or 3.Repeated Measure Experiment, or

Related/Paired Sample t-test Related/Paired Sample t-test For Directional or one tailed testsFor Directional or one tailed tests Step. 1Step. 1 HH00 : µ : µDD ≤ 0 ≤ 0 HH11 : µ : µD D > 0> 0

3636

ProblemsProblemsResearch indicates that the color red

increases men’s attraction to women (Elliot & Niesta, 2008). In the original study, men were shown women’s photographs presented on either white or red background. Photographs presented onpresented on redred were were rated significantly rated significantly more attractive more attractive than the same photographs mounted than the same photographs mounted on whit. on whit. 3737

ProblemsProblems

In a similar study, a researcher In a similar study, a researcher prepares prepares a set of 30 women’s a set of 30 women’s photographs, photographs, withwith 15 mounted on 15 mounted on a white a white background and background and 15 15 mounted on red. mounted on red. One picture is One picture is identified as the test photograph, identified as the test photograph, and appears twice in the set, once and appears twice in the set, once on white and once on red.on white and once on red.

3838

ProblemsProblems Each male participant looks through the entire Each male participant looks through the entire

set of photographs and rated the attractiveness set of photographs and rated the attractiveness of each woman on a of each woman on a 12-point scale12-point scale. The data . The data in the next slide summarizes the responses for in the next slide summarizes the responses for a sample of a sample of n=9n=9 men.men.

Set the level of significance at Set the level of significance at αα=.01=.01

for two tailsfor two tails

Do the data indicate that the color Do the data indicate that the color red red increases men’s attraction to increases men’s attraction to womenwomen ?? ??

3939

ProblemsProblemsParticipants White background Participants White background X1X1 Red Background Red Background X2 X2 DD=X2-X1 =X2-X1 DD² ²

A 6 9 +3 9A 6 9 +3 9

B 8 9 +1 1B 8 9 +1 1

C 7 10 +3 9C 7 10 +3 9

D 7 11 +4 16D 7 11 +4 16

E 8 11 +3 9E 8 11 +3 9

F 6 9 +3 9F 6 9 +3 9

G 5 11 +6 36G 5 11 +6 36

H 10 11 +1 1H 10 11 +1 1

I 8 11 +3 9 I 8 11 +3 9

ΣΣD =27 D =27 ΣΣDD²=99²=99

MMD =D =

4040

D

n

Null HypothesisNull Hypothesis For Non-Directional or two tailed testsFor Non-Directional or two tailed tests

Step. 1Step. 1HH00 : µ : µDD = 0 = 0HH11 : µ : µDD ≠ 0 ≠ 0

4141

ProblemsProblemsOne technique to help people deal with phobia is One technique to help people deal with phobia is

to have them counteract the feared objects by to have them counteract the feared objects by using using imaginationimagination to move themselves to a to move themselves to a place of safety. In an experiment test of this place of safety. In an experiment test of this technique, patients sit in front of a screen and technique, patients sit in front of a screen and are instructed to relax. Then they are shown a are instructed to relax. Then they are shown a slide of the feared object for example, a picture slide of the feared object for example, a picture of a spider, (arachnophobia). The patient of a spider, (arachnophobia). The patient signals the researcher as soon as feelings of signals the researcher as soon as feelings of anxiety begin to arise, and the researcher anxiety begin to arise, and the researcher records the amount of time that the patient was records the amount of time that the patient was able to endure looking at the slide. able to endure looking at the slide. 4242

ProblemsProblems The patient then spends two minutes The patient then spends two minutes

imagining a “safe scene” imagining a “safe scene” such as a such as a tropical tropical beachbeach (next slide) (next slide) beforebefore the slide is presented the slide is presented again. If patients can tolerate the feared object again. If patients can tolerate the feared object longer longer afterafter the imagination exercise, it is the imagination exercise, it is viewed as a reduction in the phobia. The data viewed as a reduction in the phobia. The data in next slide summarize the items recorded in next slide summarize the items recorded from a sample of from a sample of n=7n=7 patients. patients. Do the data Do the data indicate that the imagination technique indicate that the imagination technique effectively alters phobia? effectively alters phobia? .. Set the level of Set the level of significance at significance at αα=.05 for one tailed test.=.05 for one tailed test.

4343

ProblemsProblemsParticipant Before imagination Participant Before imagination X1 X1 After Imagination After Imagination X2 X2 DD=X2-X1 =X2-X1 D D² ²

A 15 24 +9 81A 15 24 +9 81

B 10 23 +13 169B 10 23 +13 169

C 7 11 +4 16C 7 11 +4 16

D 18 25 +7 49D 18 25 +7 49

E 5 14 +9 81E 5 14 +9 81

F 9 14 +5 25F 9 14 +5 25

G 12 21 +9 81G 12 21 +9 81

ΣΣD =56 D =56 ΣΣDD²=502²=502

MMD =D =

4545

D

n

Null HypothesisNull Hypothesis

For Directional or one tailed testsFor Directional or one tailed tests

Step. 1Step. 1 HH00 : µ : µD D ≤ 0 ≤ 0 (The amount of time is not increased.)(The amount of time is not increased.)

HH11 : µ : µDD > 0 > 0 (The amount of time is increased.)(The amount of time is increased.)

4646

Relationship of Statistical Relationship of Statistical TestsTests

Does this Diagram Make Sense to You?Does this Diagram Make Sense to You?

Structural Equation Modeling

Structural Model

Multiple Regression

ANOVA

t ratio

Correlation

Measurement Model

Confirmatory Factor Analysis

Exploratory Factor Analysis

EstimationEstimation The inferential process of using sample The inferential process of using sample

statistics to statistics to estimate estimate population parameter population parameter is called is called estimation.estimation.

We use estimation 1. After hypotheses test We use estimation 1. After hypotheses test when when HH0 0 is rejected. 2. when we know there is rejected. 2. when we know there is an effect and simply want to find out is an effect and simply want to find out how how much. much. 3. When we want some basic 3. When we want some basic information about an unknown population.information about an unknown population.

See the logic behind hypothesis tests See the logic behind hypothesis tests ans estimation.ans estimation.

4848

EstimationEstimation Point Estimate:Point Estimate: Interval Estimate:Interval Estimate: µ= M ± tsµ= M ± tsMM

µµ11- µ- µ 2 2 = M= M11-M-M22 ± ts(M ± ts(M11-M-M22))

µµD D = M= MDD ± ts ± tsMMDD

4949

ANANALYSIS ALYSIS OOF F VAVARIANCERIANCE ANOVAANOVA

TESTS FOR DIFFERENCES AMONG TWO TESTS FOR DIFFERENCES AMONG TWO OR MORE POPULATION MEANSOR MORE POPULATION MEANS

σσ²²=S=S²²=MS=MS

MS=Mean Squared DeviationMS=Mean Squared Deviation Ex of ANOVA Research: The effect of Ex of ANOVA Research: The effect of

temperaturetemperature on on recallrecall. .

5151

StatisticsStatisticsStandard Deviations and VariancesStandard Deviations and Variances

X X σ² = ss/N Popσ² = ss/N Pop 1 1 σ = √σ = √ss/Nss/N 22 4 s = √ss/df 4 s = √ss/df SampleSample 5 5 s² = ss/n-1 or s² = ss/n-1 or ss/dfss/df

MSMS = = SS/dfSS/df

5252

Effects of Temperature (IV) on Recall (DV)Effects of Temperature (IV) on Recall (DV)

5353

MSMS betbet = SS = SS betbet / df / df betbet

MS MS withwith = SS = SS withwith /df /df withwith

5656

ANOVAANOVA

SS SS betbet = =ΣΣ(T²/n-G²/n)(T²/n-G²/n) SS SS withwith = =ΣΣssss SS SS totaltotal =SS =SS betbet + SS + SS withwith df df betbet = K-1 = K-1 df df withwith =N-K =N-K df df totaltotal= df = df betbet + df + df withwith

5858

Post Hoc Tests (Post Tests)Post Hoc Tests (Post Tests)

Post Hoc Tests are Post Hoc Tests are additional additional hypothesis hypothesis tests that are tests that are done done after after an ANOVA to an ANOVA to determine exactly determine exactly which which mean mean difference are difference are significant and which are not.significant and which are not.

5959

Post Hoc Tests (Post Tests)Post Hoc Tests (Post Tests)

Tukey’s Honestly Significant Tukey’s Honestly Significant Difference(HSD) Test Difference(HSD) Test

HSD= qHSD= q

6060

with / nMS

ProblemsProblems The data in next slide were obtained from an The data in next slide were obtained from an

independent-measures experiment designed to independent-measures experiment designed to examine people’s performances for viewing examine people’s performances for viewing distance distance of a 60-inch high definition television. of a 60-inch high definition television. Four viewing Four viewing distances were evaluated, distances were evaluated, 99 feet, feet, 1212 feet, feet, 15 15 feet, and feet, and 1818 feet, with a separate feet, with a separate group of participants tested at each distance. group of participants tested at each distance. Each individual watched a 30-minute television Each individual watched a 30-minute television program from a specific distance and then program from a specific distance and then completed a brief questionnaire measuring their completed a brief questionnaire measuring their satisfaction with the experience. satisfaction with the experience.

6161

ProblemsProblems One question asked them to One question asked them to rate the viewing rate the viewing

distance on a scale from distance on a scale from 1 (Very Bad 1 (Very Bad definitely definitely need to move closer or farther away) to need to move closer or farther away) to 7 7 (Excellent(Excellent-perfect-perfect viewing distance). The viewing distance). The purpose of the ANOVA purpose of the ANOVA is to determine is to determine whether there are any significant differences whether there are any significant differences among the four viewing distances among the four viewing distances that were that were tested. Before we begin the hypothesis test, tested. Before we begin the hypothesis test, note that we have already computed several note that we have already computed several summary statistics for the data in next slide. summary statistics for the data in next slide. Specifically, the tretment Specifically, the tretment totals (T) totals (T) and and SS SS values are shown for the entire set of data. values are shown for the entire set of data. 6262

ProblemsProblems9 feet 9 feet 12 feet 12 feet 15 fet 15 fet 18 feet18 feet

3 3 44 7 7 6 6 NN=20=20

0 0 33 6 6 33 GG=60=60

2 2 11 5 5 4 4 ΣΣX² X² =262=262

0 0 11 4 4 33 K=K=44

0 0 11 3 3 44

TT11=5 =5 TT22=10=10 T T33=25 =25 TT44=20=20

SSSS11=8 =8 SSSS22= 8 = 8 SSSS33=10 =10 SSSS44=6=6

MM11=1 =1 MM22=2=2 M M33=5 =5 MM44=4=4

nn11=5 =5 nn22=5=5 n n33=5 =5 nn44=5=5 6363

ProblemsProblems Having these summary values simplifies the Having these summary values simplifies the

computations in the hypothesis test, and we computations in the hypothesis test, and we suggest that you always compute these suggest that you always compute these summary statistics before you begin an summary statistics before you begin an ANOVA. ANOVA.

Step 1) Step 1)

HH00 : µ : µ11== µµ22=µ=µ33=µ=µ44 (There is no treatment effect.)(There is no treatment effect.)

HH11 : : (At least one of the treatment means is different.)(At least one of the treatment means is different.)

We will set alpha at We will set alpha at α α =.05=.056464

Step 2Step 2

6565

ProblemsProblems A human factor psychologist A human factor psychologist

studied studied three computer keyboard three computer keyboard designs. designs. Three samples of Three samples of individuals were given material to individuals were given material to type on a particular keyboard, and type on a particular keyboard, and the number of errors committed the number of errors committed by each participant was recorded. by each participant was recorded. The data are on next slide.The data are on next slide.

6666

ProblemsProblemsKeyboard A Keyboard A Keyboard B Keyboard B Keyboard CKeyboard C

0 0 66 6 6 NN=15=15

4 4 8 8 5 5 GG=60=60

0 0 55 9 9 ΣΣX² X² =356=356

1 1 44 4 4

0 0 22 6 6

TT11=5 =5 TT22=25=25 T T33=30=30

SSSS11=12 =12 SSSS22=20 =20 SS SS33=14=14

MM11=1 =1 MM22=5=5 M M33=6=6 Is there a significant Is there a significant differences among the three computer keyboard designs ?differences among the three computer keyboard designs ?

ProblemsProblems Step 1) Step 1)

HH00 : µ : µ11== µµ22=µ=µ33 (No differences between the (No differences between the

computer keyboard designs )computer keyboard designs )

HH11 : : (At least one of the computer keyboard designs is (At least one of the computer keyboard designs is

different.)different.)


6868

Step 2Step 2

6969

2-WAY ANOVA2-WAY ANOVA

7070

Correlation Correlation

& &

RegressionRegression

7171

X

Y

CorrelationCorrelation

Correlation measures the Correlation measures the strengthstrength and and the the directiondirection of the relationship of the relationship between two or more variables.between two or more variables.

A correlation has three components:A correlation has three components:– The The strengthstrength of the coefficient of the coefficient– The The directiondirection of the relationship of the relationship– The The formform of the relationship of the relationship

The The strengthstrength of the coefficient of the coefficient is is indicated by the absolute value of the indicated by the absolute value of the coefficient.coefficient.– The closer the value is to The closer the value is to 1.01.0, either , either

positive or negative, positive or negative, the stronger or more the stronger or more linear the relationship.linear the relationship.

– The closer the value is toThe closer the value is to 0 0, , the weaker or the weaker or nonlinearnonlinear the relationship. the relationship.

7272

X

Y

CorrelationCorrelation

The The directiondirection of coefficient of coefficient is indicated by is indicated by the sign of the correlation coefficient.the sign of the correlation coefficient.– A A positivepositive coefficient indicates that as one coefficient indicates that as one

variable (X) increases, so does the other (Y).variable (X) increases, so does the other (Y).– AA negativenegative coefficient indicates that as one coefficient indicates that as one

variable (X) increases, the other variable (Y) variable (X) increases, the other variable (Y) decreases.decreases.

– The The form form of the relationshipof the relationship The form of the relationship is The form of the relationship is linear.linear.

In correlation variables are In correlation variables are not identified as not identified as independent or dependent independent or dependent because the because the researcher is measuring the one researcher is measuring the one relationship that is mutually shared between relationship that is mutually shared between the two variablesthe two variables– As a result, causality should not be implied with As a result, causality should not be implied with

correlationcorrelation..

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CorrelationCorrelation Remember, the correlation coefficient can only Remember, the correlation coefficient can only

measure a measure a linear relationshiplinear relationship.. A zero correlation indicates A zero correlation indicates no linear relationshipno linear relationship..

However, does not indicate no relationshipHowever, does not indicate no relationship..

a coefficient of zero rules out a coefficient of zero rules out linear linear relationship, relationship,

but a but a curvilinearcurvilinear could still exist.could still exist.– The scatterplots below illustrate this point:The scatterplots below illustrate this point:

05

1015

200

5

10

15

20

No Relationshipr = .0

05

1015

200

5

10

15

20

No Linear Relationshipr = 0

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The Correlation is based on a Statistic The Correlation is based on a Statistic Called Called CovarianceCovariance

Variance and Covariance Variance and Covariance are used to are used to measure the quality of measure the quality of an item in a test.an item in a test.

Reliability and validity measure the quality of Reliability and validity measure the quality of the entire test.the entire test.

σ²=SS/Nσ²=SS/N used for one set of data used for one set of data

VarianceVariance is the degree of variability is the degree of variability

of scores from meanof scores from mean..

The Correlational MethodThe Correlational MethodSS,SS, Standard Deviations and VariancesStandard Deviations and Variances

X X σ² = ss/N Pop σ² = ss/N Pop 1 1 σ = √σ = √ss/Nss/N 22 4 4 s = √ss/df s = √ss/df 5 5 s² = ss/n-1 or ss/df Sample s² = ss/n-1 or ss/df Sample



Sum Sum of of SquaredSquared DeviationDeviation from from MeanMean7676

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VarianceVariance

X X σ² = ss/N Popσ² = ss/N Pop 1 1 s² = ss/n-1 or ss/df Samples² = ss/n-1 or ss/df Sample 2 2 4 4 55



Sum Sum of of SquaredSquared DeviationDeviation from from MeanMean

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CovarianceCovariance CorrelationCorrelation is based on a statistic called is based on a statistic called

CovarianceCovariance (Cov xy or S xy) ….. (Cov xy or S xy) ….. r=sp/√ssx.ssyr=sp/√ssx.ssy

Covariance is a number that reflects the Covariance is a number that reflects the degree to which 2 variables vary together. degree to which 2 variables vary together.

Original DataOriginal Data X YX Y 1 31 3 2 62 6 4 44 4 5 75 7

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CovarianceCovariance

COVxy=SP/N-1COVxy=SP/N-12 ways to calculate the SP2 ways to calculate the SPSP= Σxy-(Σx.Σy)/N SP= Σxy-(Σx.Σy)/N ComputationComputation

SP= (x-μx)(y-μy) SP= (x-μx)(y-μy) DefinitionDefinition

SP requires 2 sets of dataSP requires 2 sets of dataSS requires only one set of SS requires only one set of datadata

The Correlational MethodThe Correlational Method

Correlational data can be graphed and a Correlational data can be graphed and a “line of best fit” “line of best fit” can be drawncan be drawn

1- Pearson 1- Pearson CorrelationsCorrelations

2-Spearman2-Spearman

3-Point-Biserial Correlation3-Point-Biserial Correlation

4- Partial Correlation4- Partial Correlation

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Types of CorrelationTypes of Correlation In correlational research we use continues In correlational research we use continues

variables (variables (interval or ratio scale) interval or ratio scale) for for Pearson Pearson Correlation (for linear relationship).Correlation (for linear relationship).

If it is difficult to measure a variable on an If it is difficult to measure a variable on an interval or ratio scale then we use interval or ratio scale then we use SpearmanSpearman Correlation Correlation

SpearmanSpearman Correlation Correlation uses ordinal or rank uses ordinal or rank ordered dataordered data

Spearman Correlation measures the Spearman Correlation measures the consistency of a relationship (Monotonic consistency of a relationship (Monotonic Relationship)Relationship). . Ex. nextEx. next

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MonotonicMonotonic Transformation Transformation

They are They are rank ordered rank ordered numbers numbers (DATA), and use (DATA), and use ordinal scale(data) ordinal scale(data) examples; 1, 2, 3, 4, or 2, 4, 6, 8, 10 examples; 1, 2, 3, 4, or 2, 4, 6, 8, 10 Spearman Correlation Spearman Correlation can be used to can be used to measure the measure the degree of Monotonic degree of Monotonic relationship relationship between two variables.between two variables.

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Ex. of Monotonic data Ex. of Monotonic data

X X YY

2222 8787

2525 102102

1919 1010

6 6 5 5

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Types of CorrelationTypes of Correlation Ex. A teacher may feel confident about rank Ex. A teacher may feel confident about rank

ordering students’ leadership abilities but would ordering students’ leadership abilities but would find it difficult to measure leadership on some find it difficult to measure leadership on some other scale.other scale.

The Point-Biserial Correlation.The Point-Biserial Correlation. However, we can use both continues and However, we can use both continues and

discrete variables(data) in discrete variables(data) in The Point-Biserial The Point-Biserial Correlation. (can be a substitute for two Correlation. (can be a substitute for two independent t-test)independent t-test)

In special situations we can use In special situations we can use Partial Partial Correlations.Correlations.

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The Point-Biserial CorrelationThe Point-Biserial Correlation The point-biserial correlation is used to The point-biserial correlation is used to

measure the relationship between two measure the relationship between two variables in situations in which one variable variables in situations in which one variable consist of regular, consist of regular, numerical scores numerical scores (non-(non-dichotomies), but the second variable has only dichotomies), but the second variable has only two values two values (dichotomies).(dichotomies).

We can calculate the correlation from t-testWe can calculate the correlation from t-test r² = r² = Coefficient of Determination Coefficient of Determination which which

measures the measures the effect size=deffect size=d r² = t²/t²+dfr² = t²/t²+df r = √r²r = √r² 8585


CorrelationCorrelation is the degree to which events or is the degree to which events or characteristics vary from each othercharacteristics vary from each other– Measures the Measures the strengthstrength of a relationship of a relationship– Does not Does not imply cause and effectimply cause and effect

The people chosen for a study are its The people chosen for a study are its subjects or participantssubjects or participants, collectively called a , collectively called a samplesample– The sample must be representativeThe sample must be representative

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A Partial CorrelationA Partial Correlation

Measures the relationship Measures the relationship between two variables while between two variables while controlling the influence of a controlling the influence of a third variable by holding it third variable by holding it constant.constant.

Ex. The correlation between Ex. The correlation between churches and crime. churches and crime.

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Correlational data Correlational data can be can be graphedgraphed and a and a “line of best “line of best fit” fit” can be drawncan be drawn

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Positive Correlation Positive Correlation

Positive correlation: Positive correlation: variables change in variables change in the same directionthe same direction

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Positive Correlation Positive Correlation

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Negative Correlation Negative Correlation

Negative correlation:Negative correlation: variables change in variables change in the opposite the opposite direction direction

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Negative CorrelationNegative Correlation

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No CorrelationNo Correlation

Unrelated:Unrelated: no no consistent consistent relationshiprelationship

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No CorrelationNo Correlation

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The magnitude (The magnitude (strengthstrength) of a ) of a correlation is also importantcorrelation is also important–High magnitude =High magnitude = variables which vary variables which vary

closely together; fall close to the line of closely together; fall close to the line of best fitbest fit

–Low magnitude =Low magnitude = variables which do variables which do not vary as closely together; loosely not vary as closely together; loosely scattered around the line of best fitscattered around the line of best fit

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Direction and magnitude of a correlation are Direction and magnitude of a correlation are often calculated statisticallyoften calculated statistically– Called the Called the “correlation coefficient,” “correlation coefficient,” symbolized by the symbolized by the

letter letter “r”“r” Sign (+ or -) indicates directionSign (+ or -) indicates direction Number (from 0.00 to 1.00) indicates magnitudeNumber (from 0.00 to 1.00) indicates magnitude

0.00 = no consistent relationship0.00 = no consistent relationship +1.00 = +1.00 = perfectperfect positive correlation positive correlation -1.00 = -1.00 = perfectperfect negative correlation negative correlation

Most correlations found in psychological Most correlations found in psychological research fall far short of research fall far short of “perfect”“perfect”

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Correlations can be trusted based on Correlations can be trusted based on statistical probabilitystatistical probability– ““Statistical significance”Statistical significance” means that the finding means that the finding

is unlikely to have occurred by chanceis unlikely to have occurred by chance By convention, if there is less than a 5% probability By convention, if there is less than a 5% probability

that findings are due to chance (that findings are due to chance (pp < 0.05), < 0.05), results are results are considered considered “significant” “significant” and thought to reflect the and thought to reflect the larger populationlarger population

– Generally, Generally, confidenceconfidence increases with the increases with the size of size of the sample the sample and the and the magnitude of the correlationmagnitude of the correlation

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Advantages of correlational studies:Advantages of correlational studies:

– Have high Have high external validityexternal validityCan generalize findingsCan generalize findings

– Can repeat (replicate) studies on other Can repeat (replicate) studies on other samplessamples

Difficulties with correlational studies:Difficulties with correlational studies:

– Lack Lack internal validityinternal validityResults describe but Results describe but do not do not explainexplain a a

relationshiprelationship9898

External & Internal ValidityExternal & Internal Validity

External ValidityExternal Validity

External validity addresses the ability to generalize External validity addresses the ability to generalize your study to other people and other situations.your study to other people and other situations.

Internal ValidityInternal ValidityInternal validity addresses the "true" Internal validity addresses the "true" causescauses of the of the

outcomes that you observed in your study. Strong outcomes that you observed in your study. Strong internal validity means that you not only have internal validity means that you not only have reliable measures reliable measures of your of your independent (predictors)independent (predictors) and and dependent variables (criterions) dependent variables (criterions) BUT a strong BUT a strong justification that justification that causallycausally linkslinks your independent your independent variables variables (IV) (IV) to your dependent variablesto your dependent variables (DV). (DV). 9999

The Correlational MethodThe Correlational MethodPearsonPearson

r=sp/√ssx.ssyr=sp/√ssx.ssy Original DataOriginal Data X YX Y 1 31 3 2 62 6 4 44 4 5 75 7

SPSP requires 2 sets of data requires 2 sets of dataSSSS requires only one set of data requires only one set of data

df=n-2df=n-2100100

The Correlational MethodThe Correlational MethodSpearmanSpearman

r=sp/√ssx.ssyr=sp/√ssx.ssy Original Data Original Data Ranks Ranks X Y X YX Y X Y 1 3 1 11 3 1 1 2 6 2 3 2 6 2 3 4 4 3 24 4 3 2 5 7 4 45 7 4 4

SPSP requires 2 sets of data requires 2 sets of dataSSSS requires only one set of data requires only one set of data

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Percentage of Variance Accounted for Percentage of Variance Accounted for by the Treatment (similar to Cohen’s d) by the Treatment (similar to Cohen’s d) is known as is known as ωω² Omega Squared ² Omega Squared also is also is called called Coefficient of DeterminationCoefficient of Determination

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22

2

tr

t df

Coefficient of DeterminationCoefficient of Determination

If r = 0.80 then, r ² = 0.64If r = 0.80 then, r ² = 0.64This means 64% of the This means 64% of the

variability in the Y scores can variability in the Y scores can be predicted from the be predicted from the relationship with X.relationship with X.

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ProblemsProblems Test the hypothesis for the following n=4 pairs of Test the hypothesis for the following n=4 pairs of

scores for a correlation. scores for a correlation. r=sp/√ssx.ssyr=sp/√ssx.ssy Original DataOriginal Data X YX Y 1 31 3 2 62 6 4 44 4 5 75 7

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ProblemsProblems

Step 1) Step 1) HH00 : : ρρ=0 =0 (There is no population correlation.)(There is no population correlation.)

HH11 : : ρ≠ρ≠0 0 (There is a real correlation.)(There is a real correlation.)

ΡΡ: probability or chances are…: probability or chances are…


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STEP 2STEP 2

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ProblemsProblems Test the hypothesis for the following set of n=5 Test the hypothesis for the following set of n=5

pairs of scores for a pairs of scores for a positive correlation. positive correlation. Original DataOriginal Data X YX Y 0 20 2 10 610 6 4 24 2 8 48 4 8 68 6

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ProblemsProblems

Step 1) Step 1) HH00 : : ρρ≤0 ≤0 ((The population correlation is ((The population correlation is not positive.)not positive.)

HH11 : : ρρ>0 >0 (The population correlation is (The population correlation is positivepositive.).)

ΡΡ: probability or chances are…: probability or chances are…


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Bi-Variate Regression AnalysisBi-Variate Regression Analysis Bi-variate regression Bi-variate regression analysis extends analysis extends

correlation and attempts to measure the extent correlation and attempts to measure the extent to which a to which a predictorpredictor variable (X) can be used to variable (X) can be used to make a make a prediction prediction about a about a criterioncriterion measure measure (Y). (Y).

X Y

Bi-variateBi-variate regression uses a linear linear model model to predictpredict the criterioncriterion measure.The formula for the predicted score is:

Y' = a + bXY' = a + bX 110110

Bivariate RegressionBivariate Regression The components of the line of best The components of the line of best

fit (Y' = a + bX) are:fit (Y' = a + bX) are:–the the YY-intercept -intercept ((aa) ) ConstantConstant–the slope the slope (b)(b)–Variable Variable (X)(X)

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Bivariate RegressionBivariate Regression The The Y-interceptY-intercept is the average value of Y is the average value of Y

whenwhen X is zero.X is zero.–The The Y-interceptY-intercept is also called is also called constant.constant.–Because, this is the amount of Y that is Because, this is the amount of Y that is

constant or present when the influence constant or present when the influence of X is null (0).of X is null (0).

The slope The slope is average value of a is average value of a one unit one unit change in Y change in Y for a corresponding for a corresponding one unit one unit change in X.change in X.–Thus, the Thus, the slope represents the direction slope represents the direction

and intensity of the line.and intensity of the line.112112

Regression and PredictionRegression and Prediction

Y=bX+aY=bX+aRegression LineRegression Line

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http://en.wikipedia.org/wiki/File:Linear_regression.png

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Bivariate RegressionBivariate Regression Line of Best Fit:Line of Best Fit: Y' = 2.635 + .204X Y' = 2.635 + .204X

With this equation a predicted score may be made for With this equation a predicted score may be made for any value ofany value of X X within the range of data.within the range of data.

a=2.635 a=2.635 and and b=.204b=.204

01

23

4

High School Grade Point Average

0

1

2

3

4

0

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First Year Grade Point Average

Y-intercept

2.635

Slope = .204

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Multiple Regression AnalysisMultiple Regression Analysis Multiple regression analysis is Multiple regression analysis is

an extension of an extension of bi-variate bi-variate regression, regression, in whichin which severalseveral predictorpredictor variables are used to variables are used to predict predict one criterion one criterion measure measure (Y).(Y).– In general, this method is In general, this method is

considered to be considered to be advantageous; since seldom advantageous; since seldom can an outcome measure be can an outcome measure be accurately explained by one accurately explained by one predictor variablepredictor variable. .

YX 2

X 1

X 3

Y' = a + bY' = a + b11XX11 +b +b22XX22 +b+b33XX33

E

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Relationship of Statistical Relationship of Statistical TestsTests

Does this diagram make sense to you?Does this diagram make sense to you?

Structural Equation Modeling

Structural Model

Multiple Regression

ANOVA

t ratio

Correlation

Measurement Model

Confirmatory Factor Analysis

Exploratory Factor Analysis

PARAMETRIC PARAMETRIC AND NONPARAMETRIC AND NONPARAMETRIC STATISTICAL TESTSSTATISTICAL TESTS

Parametric tests are more accurate and Parametric tests are more accurate and have 3 assumptions:have 3 assumptions:

1. Random selection1. Random selection 2. Independent of observation2. Independent of observation 3. Sample is taken from a 3. Sample is taken from a normal normal

population population with a normal distribution

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NONPARAMETRIC NONPARAMETRIC STATISTICAL TESTSSTATISTICAL TESTS

CHI SQURE: CHI SQURE: It It is like frequency distributionis like frequency distribution

Is is used for comparative studies.Is is used for comparative studies. Ex. Of the two leading brands of cola, Ex. Of the two leading brands of cola,

which is preferred by most American?which is preferred by most American?

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CHI SQURE CHI SQURE

= = ΣΣ(fo-fe)²(fo-fe)² /fe/fe

df = C-1 df = C-1

fe = n/c fe = n/c

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CHI SQURE CHI SQURE

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df=C-1df=C-1

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ProblemsProblems A psychologist examining art appreciation selected an A psychologist examining art appreciation selected an

abstract painting that had no obvious top or bottom. abstract painting that had no obvious top or bottom. Hangers were placed on the painting so that it could Hangers were placed on the painting so that it could be hung with any one of the four sides at the top. The be hung with any one of the four sides at the top. The painting was shown to a sample of n=50 participants, painting was shown to a sample of n=50 participants, and each was asked to hung the painting in the and each was asked to hung the painting in the orientation that looked correct. The following data orientation that looked correct. The following data indicate how many people choose each of the four indicate how many people choose each of the four sides to be placed at the top. sides to be placed at the top.

fofo

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Top up (correct)18

Bottom up17

Left side up7

Right side up8

ProblemsProblems The question for the hypothesis test is The question for the hypothesis test is

whether there are any preferences among whether there are any preferences among the four possible orientations. Are any of the the four possible orientations. Are any of the orientations selected more (or less) often orientations selected more (or less) often than would be expected simply by chance?than would be expected simply by chance?


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ProblemsProblems

Step 1) Step 1) HH00 : fo=fe : fo=fe no preference for any specific orientationno preference for any specific orientation

HH11 : fo≠fe : fo≠fe preference for specific orientationpreference for specific orientation

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Step 2 Step 2 df=C-1df=C-1

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t-Static t-test is used to test hypothesis about an unknown population mean, µ, when the value of...

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