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Chapter 10

The t Test for Two

Independent Samples

PowerPoint Lecture Slides

Essentials of Statistics for the Behavioral Sciences

Tenth Edition

by Frederick J Gravetter, Larry B. Wallnau, Lori-Ann B. Forzano, and James E. Witnauer

Chapter 10 Learning Outcomes

•

•

•

•

Understand structure of research study appropriate

for independent-measures t hypothesis test

Test the difference between two populations or two

treatments using independent-measures t statistic

Evaluate the magnitude of the observed mean

difference (effect size) using Cohen’s d, r2, and/or a

confidence interval

Understand how to evaluate the assumptions

underlying this test and how to adjust calculations

when needed

Tools You Will Need

•

•

•

Sample variance (Chapter 4)

Standard error formulas (Chapter 7)

The t statistic (Chapter 9)

–

–

–

Distribution of t values

df for the t statistic

Estimated standard error

10-1 Introduction to the Independent

Measures Design (1 of 2)

•

Most research studies compare two (or more) sets

of data

–

–

•

Data from two completely different, independent

participant groups (an independent-measures

research design or between-subjects design)

Data from the same or related participant group(s)

(a within-subjects design or repeated-measures

research design)

The key concept is to explain the difference

between a one sample t test and an independent

samples t test.

Introduction to the Independent-Measures

Design (2 of 2)

•

•

•

Computational procedures are considerably

different for the two designs

Each design has different strengths and

weaknesses

Consequently, only between-subjects designs are

considered in this chapter; repeated-measures

research designs will be discussed in Chapter 11

Figure 10.1 The Structure of an IndependentMeasures Research Study

10-2 The Hypotheses and the IndependentMeasures t Statistic

•

Null hypothesis for independent-measures test (no

difference between the population means)

•

Alternative hypothesis for the independentmeasures test (there is a mean difference)

The Formulas for an Independent-Measures

Hypothesis Test

•

•

The basic structure of the t statistic

t = [(sample statistic) – (hypothesized population

parameter)] divided by the estimated standard error

The Estimated Standard Error

•

•

Measure of standard or average distance between

sample statistic (M1 – M2) and the population

parameter

How much difference is reasonable to expect

between two sample means if the

null hypothesis is true (Equation 10.1)?

Box 10.1 The Variability of Difference Scores

•

•

Why add sample measurement errors (squared

deviations from mean) but subtract sample means

to calculate a difference score?

The variability for the difference in scores is found

by adding the variability for each of the two

populations

Figure 10.2 Two Population Distributions

Pooled Variance

•

Equation 10.1 shows the standard error concept but

is unbiased only if the two samples are exactly the

same size ( n1 = n2)

•

Pooled variance (sp2) provides an unbiased basis

for calculating the standard error

Estimated Standard Error

•

The estimated standard error of M1 – M2 is then

calculated using the pooled variance estimate

The Final Formula and Degrees of Freedom

•

•

The independent-measures t statistic = sample

mean difference minus population mean difference,

divided by the estimated standard error

Degrees of freedom (df) for the t statistic is

df for first sample + df for second sample

Learning Check 1 (1 of 2)

•

Which combination of factors is most likely to produce a

significant value for an independent-measures t

statistic?

A. a small mean difference and small sample

variances

B. a large mean difference and large sample variances

C. a small mean difference and large sample variances

D. a large mean difference and small sample variances

Learning Check 1 – Answer (1 of 2)

•

Which combination of factors is most likely to produce a

significant value for an independent-measures t

statistic?

A. a small mean difference and small sample

variances

B. a large mean difference and large sample variances

C. a small mean difference and large sample variances

D. a large mean difference and small sample

variances

Learning Check 1 (2 of 2)

•

Decide if each of the following statements

is True or False.

•

T/F

–

•

If both samples have n = 10, the independentmeasures t statistic will have df = 19.

T/F

–

For an independent-measures t statistic, the

estimated standard error measures how much

difference is reasonable to expect between the

sample means if there is no treatment effect.

Learning Check 1 – Answers (2 of 2)

•

False

–

•

df = (n1 – 1) + (n2 – 1) = 9 + 9 = 18

True

–

This is an accurate interpretation

10-3 Hypothesis Tests with the IndependentMeasures t Statistic

•

Independent-measures hypothesis tests use the

same four steps as other hypothesis tests we’ve

discussed

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–

–

–

State the hypotheses and select the alpha level

Locate the critical region

Obtain the data & compute the test statistic

Make a decision

Figure 10.3 The Critical Region for the IndependentMeasures Hypothesis Test in Example 10.2 (df = 14; α = .05)

Directional Hypotheses and One-Tailed Tests

• Use a directional test only when predicting a specific

direction of the difference is justified

• Locate critical region in the appropriate tail

• Report the use of a one-tailed test explicitly in the

research report, for example,

t (14) = –2.67, p ≤ .01, one-tailed

Assumptions Underlying the IndependentMeasures t Formula

•

•

•

The observations within each sample must be

independent

The two populations from which the samples are

selected must be normal

The two populations from which the samples are

selected must have equal variances (called

homogeneity of variance)

Hartley’s F-Max Test

•

Test for homogeneity of variance

–

–

Large value indicates a large difference between

the sample variances

Small value (near 1.00) indicates similar sample

variances and that the homogeneity assumption is

reasonable

Pooled Variance Alternative

•

If sample information suggests violation of

homogeneity of variance assumption

–

–

Calculate standard error, as in Equation 10.1

Adjust df for the t test as given below:

10-4 Effect Size and Confidence Intervals for the

Independent-Measures t

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If the null hypothesis is rejected, the effect size

should be determined by using either

–

Cohen’s estimated d

–

or Percentage of variance explained

Confidence Intervals for Estimating μ1 – μ2

•

•

Sample mean difference M1 – M2 is used to estimate

the population mean difference

t equation is solved for the unknown parameter (μ1 – μ2)

Confidence Intervals and Hypothesis Tests

•

•

•

•

Estimation can provide an indication of the size of

the treatment effect

Estimation can provide an indication of the

significance of the effect

If the interval contains zero, then it is not a

significant effect

If the interval does not contain zero, the treatment

effect was significant

Figure 10.4 The 95% Confidence Interval for the

Population Mean Difference from Example 10.6

Reporting the Results of an IndependentMeasures t Test

•

•

•

•

•

Report whether the difference between the two

groups was significant

Report descriptive statistics (M and SD) for each

group

Report t statistic and df

Report p-value

Report CI immediately after t, for example, 95% CI

[0.782, 7.218]

10-5 The Role of Sample Variance and Sample

Size in the Independent-Measures t Test

•

•

•

The standard error is directly related to sample

variance

Because larger variance leads to a larger error, it

also leads to a smaller value of t (closer to zero)

and reduces the likelihood of finding a significant

result

Because larger samples produce smaller error,

larger samples lead to a larger value for the t

statistic

Figure 10.5 Two Sample Distributions

Representing Two Different Treatments

Figure 10.6 Two Sample Distributions

Representing Two Different Treatments

Learning Check 2 (1 of 2)

•

For an independent-measures research study, the

value of Cohen’s d or r2 helps to describe _____.

A. the risk of a Type I error

B. the risk of a Type II error

C. how much difference there is between the

two treatments

D. whether the difference between the two

treatments is likely to have occurred by

chance

Learning Check 2 – Answer (1 of 2)

•

For an independent-measures research study, the

value of Cohen’s d or r2 helps to describe _____.

A. the risk of a Type I error

B. the risk of a Type II error

C. how much difference there is between

the two treatments

D. whether the difference between the two

treatments is likely to have occurred by

chance

Learning Check 2 (2 of 2)

•

Decide if each of the following statements

is True or False.

•

T/F

–

•

The homogeneity assumption requires the two

sample variances to be equal

T/F

–

If a researcher reports that t(6) = 1.98, p > .05, then

H0 was rejected

Learning Check 2 – Answers (2 of 2)

•

False

–

•

The assumption requires equal population

variances but the test is valid if sample variances

are similar

False

–

H0 is rejected when p the critical

value of t

SPSS Output for the Independent-Measures

Hypothesis Test for Example 10.2

Group Statistics

VAR00002

VAR00001

N

Mean

Std. Deviation

Std. Error Mean

1.00

8

8.0000

2.92770

1.03510

2.00

8

12.0000

3.07060

1.08562

Independent Sample test

Levene’s Test for

Equality of Variances:

F

VAR00001

Equal variances

assumed

Levene’s Test for

Equality of Variances:

Sig.

.000

1.000

Equal variances

not assumed

Test for

Equality of

Means: t

Test for Equality of

Means: df

−2.667

14

−2.667

13.968

Independent Sample test

Test for

Equality of

Means: Sig. (2tailed)

VAR00001

Test for

Equality of

Means: Means

Difference

Test for Equality

of Means: Std.

Error Difference

Test for Equality of

Means: 95%

Confidence Interval of

the Difference: Lower

Test for Equality of Means:

95% Confidence Interval of

the Difference: Upper

Equal variances

assumed

.018

−4.00000

1.50000

−7.21718

−.78282

Equal variances not

assumed

.018

−4.00000

1.50000

−7.21786

−.78214

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