Chapter 9

HYPOTHESIS TESTING USING THE TWO-SAMPLE t-TEST

Going Forward

Your goals in this chapter are to learn:• The logic of a two-sample experiment• The difference between independent samples

and related samples• When and how to perform the independent-

samples t-tests• When and how to perform the related-

samples t-test• What effect size is and how it is measured

using Cohen’s d or 2pbr

Understanding the Two-Sample Experiment

Two-Sample Experiment

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

• Condition 1 produces sample mean representing

• Condition 2 produces sample mean representing

1X

2X1

2

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

• The two versions of the two-samplet-test are– 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 parametric procedure used for testing two sample means from independent samples

• Independent samples result when we randomly select participants for a condition without regard to who else has been selected for either condition

Assumptions of the IndependentSamples t-Test

• The dependent scores are normally distributed interval or ratio scores.

• The populations have homogeneous variance. Homogeneity of variance means the variance of the populations being represented are equal. )( 2

X

Statistical Hypotheses

• For a two-tailed test, the statistical hypotheses are

• H0 implies both samples represent the same population of scores

• Ha implies the means from our conditions each represent a different population of scores

0:H

0:H

21a

210

Sampling Distribution

The sampling distribution of differences between the means is the distribution of all possible differences between two means when both samples are drawn from the one raw score population that H0 says we are representing.

Performing the Independent Samples t-Test

1. Compute the mean and estimated population variance for each conditionRemember: The formula for the estimated variance in each condition is

1

)( 22

2

nnX

XsX

Performing the Independent Samples t-Test

2. Compute the pooled variance using the formula

)1()1(

)1()1(

21

222

2112

pool

nn

snsns

Performing the Independent Samples t-Test

3. Compute the standard error of the difference. This is the standard deviation of the sampling distribution of differences between means. The formula is

21

2pool

11)(

21 nnss XX

Performing the Independent Samples t-Test

4. Compute tobt for two independent samples using the formula

21

)()( 2121obt

XXs

XXt

One-Tailed Tests

The statistical hypotheses for a one-tailed test of independent samples are

OR

If 1 is expected to If 2 is expected tobe larger than 2 be larger than 1

0:H

0:H

21a

210

0:H

0:H

21a

210

One-Tailed Tests

Conduct one-tailed tests only when you can confidently predict the direction the dependent scores will change.

One-Tailed Tests

1. Decide which and corresponding is expected to be larger

2. Arbitrarily decide which condition to subtract from the other

3. Decide whether the difference will be positive or negative

4. Create Ha and H0 to match this prediction5. Locate the region of rejection6. Complete the t-test as described previously

X

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

Interpreting the Independent-Samples t-Test

• In a two-tailed t-test of independent samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit

• Otherwise, fail to reject H0

The Related Samples t-Test

Related-Samples t-Test

The related-samples t-test is 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 producing related samples are the matched-samples design and the repeated-measures design

Matched-Samples Design

• The researcher matches each participant in one condition with a particular participant in the other condition

• We do this so we have more comparable samples

Repeated-Measures Design

• Each participant is tested under both conditions of the independent variable

• That is, each participant is measured under condition 1 and again under condition 2

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

Statistical Hypotheses

The statistical hypotheses for a two-tailed related-samples t-test are

0:H

0:H

a

0

D

D

Sampling Distribution

The sampling distribution of mean differences shows all possible values of the population mean of the difference scores ( ) that occur when samples are drawn from the population of difference scores that H0 says we are representing.

D

Performing the Related-Samples t-Test

1. Compute the estimated variance of the difference scores ( ) using the formula

where N equals the number of difference scores

2Ds

1

)( 22

2

NND

DsD

2. Compute the standard error of the mean difference ( ) using the formula

Performing the Related-Samples t-Test

Ds

N

ss DD

2

3. Find tobt using the formula

Performing the Related-Samples t-Test

D

Dobt s

Dt

One-Tailed Tests

The statistical hypotheses for a one-tailed t-test of related samples are

If we expect the If we expect thedifference to be difference to belarger than 0 less than 0

0:H

0:H

a

0

D

D

0:H

0:H

a

0

D

D

Critical Values

The critical value (tcrit) is determined based on

• degrees of freedom df = N – 1 where N is the number of difference scores

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

Interpreting the Related-Samples t-Test

• In a two-tailed test of related samples, reject H0 if tobt is greater than (beyond) +tcrit or if tobt is less than (beyond) –tcrit

• Otherwise, fail to reject H0

Describing Effect Size

Effect Size

• Effect size indicates the amount of influence changing the conditions of the independent variable had on dependent scores

• The larger the effect size, the more scientifically important the independent variable is

Computing Effect Size

Cohen’s d is used to compute effect size

2 21pools

XXd

2

Ds

Dd

Independent Samplest-Test

Related Samplest-test

Interpreting Effect Size

We interpret the Cohen’s d using a small, medium, or large effect size classification•d = 0.2 is a small effect•d = 0.5 is a medium effect•d = 0.8 is a large effect

Proportion of Variance Accounted For

• The proportion of variance accounted for is the proportion of the differences in scores that can be attributed to changing the conditions in the independent variable

• We use the formula for the squared point-biserial correlation coefficient

dft

tr

2obt

2obt2

pb)(

)(

Example 1

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

Sample 1 Sample 2

14 14 13 15 11 15

13 10 12 13 14 13

14 15 17 14 14 15

Example 1

556.131 X

778.132 X

944.11 s302.12 s

91 n92 n

737.2

)19()19(

695.1)19(779.3)19(

)1()1(

)1()1(

21

222

2112

nn

snsnspool

Example 1

The standard error of the difference is

780.0)222.0)(737.2(

9

1

9

1)737.2(

11)(

21

2

21

nn

ss poolXX

Example 1

285.0780.0

222.0780.0

0)778.13556.13(

)()(

21

2121obt

XXs

XXt

Example 1

• tcrit for df = (9 – 1) + (9 – 1) = 16 with = .05 and a two-tailed test is 2.120.

• Reject H0 if tobt is greater than +2.120 or if tobt is less than –2.120.

• Because tobt of – 0.285 is not beyond the –tcrit of –2.120, it does not lie within the rejection region. We fail to reject H0.

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

14 14 13 15 16 15

13 10 12 16 14 13

14 15 17 18 17 19

Example 2

First, we find the differences between the matched scores

Sample 1 Sample 2 Differences

14 14 13 15 16 15 -1 -2 -2

13 10 12 16 14 13 -3 -4 -1

14 15 17 18 17 19 -4 -2 -2

Example 2

260.6373.0

333.2

925.1

0333.2

2obt

Ns

Dt

D

D

Example 2

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

• Reject H0 if tobt is greater than +2.306 or if tobt is less than –2.306.

• Because tobt of –6.260 is beyond the –tcrit value of –2.306, it lies within the rejection region. We reject H0.

Example 2

Effect size

087225133322.

.

.

Ds

Dd

Example 2

Proportion of varianceaccounted for

830.0188.47

188.39

826.6

26.6

)(

)(

2

2

2

22

dft

tr

obt

obtpb