t-Statict-Static 1. Single Sample or One Sample t-Test1. Single Sample or One Sample t-Test

AKA student t-test.AKA student t-test.

2. Two Independent sample 2. Two Independent sample

t-Test, AKA t-Test, AKA BetweenBetween Subject Designs or Subject Designs or Matched subjects Experiment.Matched subjects Experiment.

3. Related Samples t-test or Repeated 3. Related Samples t-test or Repeated Measures Experiment AKA Measures Experiment AKA Within Within Subject Designs or Paired Sample T-Subject Designs or Paired Sample T-Test .Test .

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CHAPTER 11CHAPTER 11RepeatedRepeated Measure Measure Experiment, Experiment, Related/Related/PairedPaired Sample Sample t-test or,t-test or,Within Subject Within Subject Experiment DesignExperiment Design

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RepeatedRepeated Measure Experiment, or Measure Experiment, or Related/PairedRelated/Paired Sample t-test Sample t-test Within Within

Subject Subject Experiment DesignExperiment DesignA 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.

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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 Repeated Measure Experiment, or Repeated Measure Experiment, or

Related/Paired Sample t-testRelated/Paired Sample t-test If non-directional or two tailed test, thenIf non-directional or two tailed test, then Step. 1Step. 1

HH00 : : µµDD = 0 = 0

HH11 : : µµDD ≠ 0 ≠ 0

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Null HypothesisNull Hypothesis t-Statistics:t-Statistics: If directional or one tailed testsIf directional or one tailed tests Step. 1Step. 1

HH00 : : µµDD ≤ 0 ≤ 0

HH11 : : µµD D > 0> 0

or, Step. 1or, Step. 1

HH00 : : µµDD ≥ 0 ≥ 0

HH11 : : µµD D < 0< 0

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None-directional None-directional Hypothesis TestHypothesis Test

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Calculations for Calculations for t-testt-testStep 3: Computations/ Calculations or Collect Step 3: Computations/ Calculations or Collect

Data and Compute Sample StatisticsData and Compute Sample Statistics

tt= = MMDD--μμDD

SSMMDD

SSMMDD= = S/√n or

SSMMDD== MMDD--μμDD

t t

MMDD=t.S=t.SMMDD++μμDD

μμDD=M=MDD- S- SMMDD..t t SSMD= MD= estimated standard error of the mean differencethe mean difference

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2 /S n

Calculations for Calculations for t-testt-testStep 3: Computations/ Calculations or Collect Step 3: Computations/ Calculations or Collect

Data and Compute Sample StatisticsData and Compute Sample Statistics

df=n-1df=n-1

Difference Score Difference Score DD== X X22-X-X11

MMD D == ΣDΣD

nn

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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 4 s = √ss/df s = √ss/df 5 5 s² = ss/n-1 or ss/df s² = ss/n-1 or ss/df SampleSample

SS=SS=ΣxΣx²-(Σx)²/n ²-(Σx)²/n new new SS= SS=ΣΣDD²-(Σ²-(ΣDD)²/n)²/n

SS=SS=ΣΣ(( x-x-μμ))²²

Sum Sum of of SquaredSquared DeviationDeviation from from MeanMean99

Cohn’s d=Cohn’s d=Effect Size for tEffect Size for tUse Use SS instead of instead of σσ for for t-testt-test

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d = Md = MDD/s/s

S= MS= MDD/d/d

MMDD= d . s= d . s

Percentage of Variance Percentage of Variance Accounted for by the Treatment Accounted for by the Treatment

(similar to Cohen’s d) Also (similar to Cohen’s d) Also known as known as ωω²² Omega Squared Omega Squared

1111

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2

tr

t df

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 whitewhite or redred background. Photographs presented onpresented on redred were were rated significantly rated significantly more attractive more attractive than than the same photographs mounted on the same photographs mounted on white. white.

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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 on whitewhite and once on and once on red.red.

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

photographs and rated the attractiveness of each photographs and rated the attractiveness of each woman on a woman on a 12-point scale12-point scale. The data in the next slide . The data in the next slide summarizes the responses for a sample of summarizes the responses for 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 the color Do the data indicate the color red red increases increases men’s attraction to womenmen’s attraction to women ??

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ProblemsProblemsParticipants Participants White background White background X1 X1 Red Background Red Background X2 X2 DD=X2-X1 =X2-X1 D²

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 =

1515

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

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ProblemsProblems One technique to help people deal with phobia One technique to help people deal with phobia

is to have them counteract the feared objects is to have them counteract the feared objects by using by 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 are instructedinstructed to relax. Then they are shown a 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. 1717

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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 the imagination technique indicate 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.

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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 =

2121

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.)

2222

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