08 two sample tests

8/6/2019 08 Two Sample Tests

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Overview

Two paired samples: Within-Subject Designs-Hypothesis test

-Confidence Interval

-Effect Size

Two independent samples: Between-Subject Designs-Hypothesis test-Confidence interval

-Effect Size


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Two kinds of studies

There are two general research strategies that can be used toobtain the two sets of data to be compared:

1. The two sets of data could come from two independent populations (e.g. women and men, or students fromsection A and from section B)

2. The two sets of data could come from related

populations (e.g. before treatment and after treatment)


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Part I

Two paired samples: Within-Subject Designs-Hypothesis test


-Effect Size


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Paired T-Test for Within-Subjects Designs

Our hypotheses:

Ho: QD = 0

HA: QD { 0

To test the null hypothesis, well again compute a tstatistic and look it up in the t table.

Paired Samples t

t = D - QD sD =s

D

sn


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Steps for Calculating a Test Statistic

Paired Samples T

1. Calculate difference scores

2. Calculate D

3. Calculate s d

4. Calculate T and d.f.

5. Use Table E.6


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Confidence Intervals for Paired Samples

Paired Samples t

D s t (s D)

General formula

X s t (SE)


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Effect Size for Dependent Samples

Paired Samples d

One Sample d

s

X d H 0

Q!

D s

Dd !


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ExerciseIn Everitts study (1994), 17 girls being treated for anorexia were weighed before and after treatment.Difference scores were calculated for each participant.

Change in Weightn = 17

= 7.26sD = 7.16

D

Test the null hypothesis that there was no change in weight.

Compute a 95% confidence interval for the mean difference.

Calculate the effect size


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ExerciseChange in Weightn = 17

= 7.26sD = 7.16

D

74.1

17

16.7!!SE

17.474.1

026.7)16( !!t

01. p

T-test


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= 7.26sD = 7.16

D

12.2!crit t

)74.1(12.226.7 s!CI

)95.10,57.3(

ConfidenceInterval


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= 7.26sD = 7.16

D

16.726.7

!d

01.1!d

Effect Size


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Part II

Two independent samples: Between-Subject Designs-Hypothesis test


-Effect Size


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T-Test for Independent SamplesThe goal of a between-subjects research study is to evaluatethe mean difference between two populations (or betweentwo treatment conditions).

Ho: Q1 = Q2

HA: Q1 { Q 2

We cant compute difference scores, so


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T-Test for Independent Samples

We can re-write these hypotheses as follows:

Ho: Q1 - Q2 = 0

HA: Q1 - Q2 { 0

To test the null hypothesis, well again compute a tstatistic and look it up in the t table.


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General t formula

t = sample statistic - hypothesized population parameter

estimated standard error

?21

! X X s

Independent samples t

21

)()( 2121

X X s X X

t ! Q Q

One Sample t

X

H test s

X t 0

Q!


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Standard Error for a Difference in Means

The single-sample standard error ( s x ) measures howmuch error expected between X and Q.

The independent-samples standard error (s x1-x2 )measures how much error is expected when you areusing a sample mean difference (X 1 X2) to represent a

population mean difference.

21 X X s


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Each of the two sample means represents its own population mean, butin each case there is some error.

The amount of error associated with each sample mean can be measured by computing the standard errors.

To calculate the total amount of error involved in using two samplemeans to approximate two population means, we will find the error fromeach sample separately and then add the two errors together.

2

22

1

21

21

n

s

n

s s X X !


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But

This formula only works when n 1 = n 2. When the twosamples are different sizes, this formula is biased .

This comes from the fact that the formula above treats thetwo sample variances equally. But we know that thestatistics obtained from large samples are better estimates,so we need to give larger sample more weight in our

estimated standard error.

2

22

1

21

21

n

s

n

s s X X !


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We are going to change the formula slightly so that we usethe pooled sample variance instead of the individual samplevariances.

This pooled variance is going to be a weighted estimate of the variance derived from the two samples.

s p2

!SS 1 SS 2

df 1 df 2

2

2

1

2

21

n

s

n

s s p p X X !


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One-Sample T

1. Calculate sample mean

2. Calculate standard error


4. Use Table D


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Independent Samples T

1. Calculate X 1-X2

2. Calculate pooled variance

3. Calculate standard error


5. Use Table E.6

s p2

!

SS 1 SS 2df 1 df 2

s p2

1

s p2

n2

t !( X X 2) ( Q Q 2 )

s x

1x

2 d.f. = (n 1 - 1) + (n 2 - 1)



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IllustrationA developmental psychologist would like to examine thedifference in verbal skills for 8-year-old boys versus 8-year-old girls. A sample of 10 boys and 10 girls isobtained, and each child is given a standardized verbal

abilities test. The data for this experiment are as follows:

Girls Boys

n1 = 10= 37

SS1 = 150 X 1

n2 = 10= 31

SS2 = 210 X 2


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STEP 1: get mean difference

621 ! X X

Girls Boys

n1

= 10= 37

SS 1 = 150 X 1

n2

= 10= 31

SS 2 = 210 X 2

Illustration


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STEP 2: Compute Pooled Variance

s p2

!SS 1 SS 2

df 1 df 2!

150 2 10

(10 1) (10 1)

!360

18

! 2 0

Girls Boys

n1

= 10= 37

SS 1 = 150 X 1

n2

= 10= 31

SS 2 = 210 X 2

Illustration


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STEP 3: Compute Standard Error

Girls Boys

n1

= 10= 37

SS 1 = 150 X 1

n2

= 10= 31

SS 2 = 210 X 2

SE ! s p

2

n 1

s p2

n 2!

20

10

20

10! 4 ! 2

Illustration


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STEP 5: Use table E.6

Girls Boys

n1 = 10= 37

SS 1 = 150 X 1

n2 = 10= 31

SS 2 = 210 X 2

T = 3 with 18 degrees of freedom

For alpha = .01, critical value of t is 2.878

Our T is more extreme, so we reject the null

There is a significant difference between boys and girls

Illustration


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SampleData

HypothesizedPopulationParameter

SampleVariance

EstimatedStandard

Error

t-statistic

Singlesample

t-statistic

Independent

samplest-statistic

X 1 X 2 Q1 Q 2 s p2

n1

s p2

n2

s p2

!

SS 1

SS 2

d f 1 d f 2

X Qs

2

n s

2 !SS f t !

X Q s x

t !( X 1 X 2 ) ( Q1 Q 2 )

s x1 x2


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Confidence Intervals for Independent Samples

One Sample t

X s t (s x)

General formula

X s t (SE)

Independent Sample t

(X1-X 2) s t (s x1-x2 )


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Effect Size for Independent Samples

One Sample d

Independent Samples d

s

X d H 0

Q!

p s X X d 21!


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Exercise

Subjects are asked to memorize 40 noun pairs. Ten subjectsare given a heuristic to help them memorize the list, theremaining ten subjects serve as the control and are given nohelp. The ten experimental subjects have a X-bar = 21 and aSS = 100. The ten control subjects have a X-bar = 19 and a SS= 120.

Test the hypothesis that the experimental group differs fromthe control group.

Give a 95% confidence interval for the difference betweengroups

Give the effect size


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ExerciseExperimental Control

n1 = 10= 21SS 1 = 100

X 1n2 = 10= 19SS 2 = 120

X 2

221 !X X

2.1218220

)110()110(120100

21

212 !!!!d f d f

SS SS s p

56.144.210

2.1210

2.12

2

2

1

2

!!!!n

s

n

sSE p p

T-test


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Exercise

28.156.1

02)()(

21

2121 !!! x x s

X X t Q Q

d.f. = (n 1 - 1) + (n 2 - 1) = (10-1) + (10-1) = 18

20." p

T-test

Experimental Control

n1 = 10= 21SS 1 = 100

X 1n2 = 10= 19SS 2 = 120

X 2


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Exercise

101.2!crit t

)56.1(101.22 s!C I

ConfidenceInterval

)28.5,28.1(


n1 = 10= 21SS 1 = 100

X 1n2 = 10= 19SS 2 = 120

X 2


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Exercise

57.!d

Effect Size

p s X X

d 21!

2.12

2!d


n1 = 10= 21SS 1 = 100

X 1n2 = 10= 19SS 2 = 120

X 2


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Summary

Hypothesis Tests

Confidence Intervals

Effect Sizes

1 Sample

2 Paired Samples

2 Independent Samples


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R eview


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SampleData

HypothesizedPopulationParameter

SampleVariance

EstimatedStandard

Error

t-statistic

One samplet-statistic

Pairedsamples t-

statistic

Independent

samplest-statistic

X

X 1 X 2

Q

Q1 Q2

s2

n

s p

2

n1

s p

2

n2

s2

!SS f

s p2

!

SS 1

SS 2

d f 1 d f 2

t ! Q

s x

t !( X 1 X 2 ) ( Q1 Q 2 )

s x1 x2

D D Q s

2

ndf

SS s D!2

D

D

s

Dt

Q!


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Confidence Intervals

Paired Samples t

D s t (s D)

One Sample t

X s t (SE)

Independent Sample t

(X1-X 2) s t (s x1-x2 )


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Effect Sizes

Paired Samples d

One Sample d

Independent Samples d

D s D

d !

s

X d 0

Q!

p s X X

d 21!

08 two sample tests

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