Chapter Twelve The Two-Sample t-Test. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 2 is the mean of the first sample is the

Chapter Twelve

The Two-Sample

t-Test

Copyright © Houghton Mifflin Company. All rights reserved. Chapter 12 - 2

• is the mean of the first sample

• is the mean of the second sample

• is the estimated population standard deviation of the first sample

• is the estimated population standard deviation of the second sample

• is the number of scores in the first sample

• is the number of scores in the second sample

1X

2X1s

2s

1n

2n

New Statistical Notation


Understanding the Two-Sample Experiment


• Participants’ scores are measured under two conditions of the independent variable

• Condition 1 produces sample mean that represents

• Condition 2 produces sample mean that represents

1X

2X1

2

Two-Sample Experiment


Two-Sample t-Test

• The parametric statistical procedure for determining whether the results of a two-sample experiment are significant is

the two-sample t-test

• There are two versions of the two-sample t-test– The independent samples t-test– The related samples t-test.


Relationship in the Population in a Two-sample Experiment


The Independent Samples t-Test


Independent Samples t-Test

• The independent samples t-test is the parametric procedure used for significance testing of two sample means from independent samples

• Two samples are independent when we randomly select and assign a participant to a sample


Assumptions of the Independent Samples t-Test

1. The dependent scores measure an interval or ratio variable

2. The populations of raw scores form normal distributions

3. The populations have homogeneous variance. Homogeneity of variance means that the variance of all populations being represented are equal.

4. While Ns may be different, they should not be massively unequal.


• Two-tailed test

• One-tailed test– If 1 is expected to If 2 is expected to

be larger than 2 be larger than 1

0:H

0:H

21a

210

0:H

0:H

21a

210

0:H

0:H

21a

210

Statistical Hypotheses


Critical Values

• Critical values for the independent

samples t-test (tcrit) are determined

based on degrees of freedom df =

(n1 - 1) + (n2 - 1), the selected , and

whether a one-tailed or two-tailed test

is used


Sampling Distribution

• The sampling distribution of

differences between means is the

distribution of all possible differences

between two means when they are

drawn from the raw score population

described by H0


1

)( 22

2

nnX

XsX

Computing the Independent Samples t-Test

1.Calculate the estimated population variance for each condition


)1()1(

)1()1(

21

222

2112

pool

nn

snsns


2.Compute the pooled variance


21

2pool

11)(

21 nnss XX


3.Compute the standard error of the difference


21

)()( 2121obt

XXs

XXt


4.Compute tobt for two independent samples


2121

222

211

2121obt

11)1()1()1()1(

)()(

nnnnsnsn

XXt


• These steps can be combined into the

following computational formula for the

independent samples t-test


)())((

)())((

21crit

2121crit

21

21

XXts

XXts

XX

XX

Confidence Interval

• When the t-test for independent samples is significant, a confidence interval for the difference between two ms should be computed


Power

• To maximize power in the independent samples t-test, you should maximize the difference produced by the two conditions

• Minimize the variability of the raw scores

• Maximize the sample ns


Related Samples t-Test


Related Samples

• The related samples t-test is the parametric inferential procedure used when we have two sample means from two related samples

• Related samples occur when we pair each score in one sample with a particular score in the other sample

• Two types of research designs that produce related samples are matched samples design and repeated measures design


Matched Samples Design

• In a matched samples design, the researcher matches each participant in one condition with a participant in the other condition

• We do this so that we have more comparable participants in the conditions


Repeated Measures Design

In a repeated measures design, each

participant is tested under all conditions of

the independent variable.


Assumptions of the Related Samples t-Test

• The assumptions of the related samples t-test are when the dependent variable involves an interval or ratio scale

• The raw score populations are at least approximately normally distributed

• The populations being represented have homogeneous variance

• Because related samples form pairs of scores, the n in the two samples must be equal


Transforming the Raw Scores

• In a related samples t-test, the raw scores are transformed by finding each difference score

• The difference score is the difference between the two raw scores in a pair

• The symbol for a difference score is D


0:H

0:H

a

0

D

D

0:H

0:H

a

0

D

D

0:H

0:H

a

0

D

D

Statistical Hypotheses

• Two-tailed test

• One-tailed testIf we expect the If we expect the

difference to be difference to be

larger than 0 less than 0


Estimated Population Variance of the Difference Scores

• The formula for the estimated population variance of the difference scores is

1

)( 22

2

NND

DsD


Standard Error of the Mean Difference

• The formula for the standard error of the mean difference is

N

ss DD

2


D

D

s

Dt

obt

Computing the Related Samples t-Test

• The computational formula for the related samples t-test is


Critical Values

• The critical value (tcrit) is determined based on degrees of freedom df = N - 1

• The selected , and whether a one-tailed or two-tailed test is used


DtsDts DDD ))(())(( critcrit

Confidence Interval

• When the t-test for related samples is significant, a confidence interval for D should be computed


Power

• The related samples t-test is intrinsically more powerful than an independent samples t-test

• To maximize the power you should– Maximize the differences in scores

between the conditions.– Minimize the variability of the scores within

each condition.– Maximize the size of N.


Describing the Relationship in a Two-Sample Experiment


Describing the Relationship

• Once a t-test has been shown to be significant, the next step is to describe the relationship

• In order to describe the relationship, you should– Compute a confidence interval– Graph the relationship– Compute the effect size– Compute the appropriate correlation coefficient to

determine the strength of the relationship


Sample Line Graphs Describing a Significant Relationship


dft

tr

2obt

2obt

pb )(

)(

Using rpb

• Because a two-sample t-test involves

one dichotomous X variable (the two

conditions of the independent variable)

and one continuous interval or ratio Y

variable, the point-biserial correlation

coefficient is the appropriate coefficient

to use


Degrees of Freedom for the rpb

• For independent samples, df = (n1 - 1) + (n2 - 1), where n is the number of scores in a sample

• For related samples, df = N - 1, where N is the number of difference scores


• In a two-sample experiment, equals the proportion of the variance accounted for

• This proportion of variance accounted for also is called the effect size in an experiment

• Effect size indicates how big a role changing the conditions of the independent variable plays in determining differences in dependent scores

2pbr

Effect Size


Sample 1 Sample 2

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example 1

• Using the following data set, conduct an independent samples t-test. Use = 0.05 and a two-tailed test.


556.131 X

778.132 X944.11 s302.12 s

91 n92 n

Example 1

•

•

•

•

•

•


285.0780.0

222.0

)222.0)(737.2(

222.0

91

91

)19()19(695.1)19(779.3)19(

0)778.13556.13(

11)1()1()1()1(

)()(

2121

222

211

2121obt

nnnnsnsn

XXt

Example 1


Example 1

• Because tobt does not lie within the

rejection region, we fail to reject H0


Sample 1 Sample 2

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example 2

• Using the following data set, conduct a

related samples t-test. Use = 0.05 and a

two-tailed test.


Sample 1 Sample 2 Differences

14 14 13 15 11 15 -1 3 -2

13 10 12 13 14 13 0 -4 -1

14 15 17 14 14 15 0 1 2

Example 2

• First, we find the differences between

the matched scores


316.0703.0

222.0

91

444.4

0222.0

1)( 2

obt

Ns

Dt

D

D

Example 2


Example 2

• Using = 0.05 and df = 8, tcrit = 2.306

• Because tobt does not lie within the rejection region, we fail to reject H0

Documents

Chapter Twelve The Two-Sample t-Test. Copyright © Houghton Mifflin Company. All rights reserved.Chapter 12 - 2 is the mean of the first sample is the