1. Chapter 10
The t Test for Two
Independent Samples
PowerPoint Lecture Slides
Essentials of Statistics for the
Behavioral Sciences
Eighth Edition
by Frederick J Gravetter and Larry B. Wallnau
2. Chapter 10 Learning Outcomes
• Understand structure of research study appropriate
for independent-measures t hypothesis test1
• Test difference between two populations or two
treatments using independent-measures t statistic2
• Evaluate magnitude of the observed mean difference
(effect size) using Cohen’s d, r2, and/or a confidence
interval
3
• Understand how to evaluate the assumptions
underlying this test and how to adjust calculations
when needed
4
3. 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
4. 10.1 Independent-Measures
Design Introduction
• Most research studies compare two (or more)
sets of data
– Data from two completely different, independent
participant groups (an independent-measures or
between-subjects design)
– Data from the same or related participant group(s)
(a within-subjects or repeated-measures design)
5. 10.1 Independent-Measures
Design Introduction (continued)
• 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 designs will be reserved for
discussion in Chapter 11
7. 10.2 Independent-Measures
Design t Statistic
• Null hypothesis for independent-measures
test
• Alternative hypothesis for the independent-
measures test
0: 210 H
0: 211 H
8. Independent-Measures
Hypothesis Test Formulas
• Basic structure of the t statistic
• t = [(sample statistic) – (hypothesized population
parameter)] divided by the estimated standard error
• Independent-measures t test
)
2121
2
)()(
M(M1
s
MM
t
9. Estimated standard error
• Measure of standard or average distance
between sample statistic (M1-M2) and the
population parameter
• How much difference it is reasonable to
expect between two sample means if the
null hypothesis is true (equation 10.1)
2
2
2
1
2
)(
1
21
n
s
n
s
s MM
10. Pooled Variance
• Equation 10.1 shows standard error concept
but is unbiased only if n1 = n2
• Pooled variance (sp
2 provides an unbiased
basis for calculating the standard error
21
212
dfdf
SSSS
sp
11. Degrees of freedom
• Degrees of freedom (df) for t statistic is
df for first sample + df for second sample
)1()1( 2121 nndfdfdf
Note: this term is the same as the denominator of the
pooled variance
12. Box 10.1
Variability of Difference Scores
• Why add sample measurement errors
(squared deviations from mean) but subtract
sample means to calculate a difference score?
14. Estimated Standard Error for the
Difference Between Two Means
2
2
1
2
)( 21
n
s
n
s
s
pp
MM
The estimated standard error of M1 – M2 is then
calculated using the pooled variance estimate
16. Learning Check
• Which combination of factors is most likely to
produce a significant value for an
independent-measures t statistic?
• a small mean difference and small sample variancesA
• a large mean difference and large sample variancesB
• a small mean difference and large sample variancesC
• a large mean difference and small sample variancesD
17. Learning Check - Answer
• Which combination of factors is most likely to
produce a significant value for an
independent-measures t statistic?
• a small mean difference and small sample variancesA
• a large mean difference and large sample variancesB
• a small mean difference and large sample variancesC
• a large mean difference and small sample variancesD
18. Learning Check
• Decide if each of the following statements
is True or False
• If both samples have n = 10, the
independent-measures 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
T/F
19. Learning Check - Answers
• df = (n1-1) + (n2-1) = 9 + 9 = 18False
• This is an accurate interpretationTrue
20. Measuring Effect Size
• If the null hypothesis is rejected, the size of
the effect should be determined using either
• Cohen’s d
• or Percentage of variance explained
2
21
deviationstandardestimated
differencemeanestimated
estimated
ps
MM
d
dft
t
r
2
2
2
22. Confidence Intervals for
Estimating μ1 – μ2
• Difference M1 – M2 is used to estimate the
population mean difference
• t equation is solved for unknown (μ1 – μ2)
)(2121 21 MMtsMM
23. 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 contain zero, then it is not a
significant effect
• If the interval does NOT contain zero, the
treatment effect was significant
25. In the Literature
• Report whether the difference between the
two groups was significant or not
• Report descriptive statistics (M and SD) for
each group
• Report t statistic and df
• Report p-value
• Report CI immediately after t, e.g., 90% CI
[6.156, 9.785]
26. Directional Hypotheses and
One-tailed Tests
• Use directional test only when predicting a
specific direction of the difference is justified
• Locate critical region in the appropriate tail
• Report use of one-tailed test explicitly in the
research report
28. Figure 10.7 Two Samples from
Different Treatment Populations
29. 10.4 Assumptions for the
Independent-Measures t-Test
• 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
– Homogeneity of variance
30. Hartley’s F-max test
• Test for homogeneity of variance
– Large value indicates large difference between
sample variance
– Small value (near 1.00) indicates similar sample
variances
(smallest)s
(largest)
max 2
2
s
F
31. Box 10.2
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:
11 2
2
2
2
2
1
2
1
2
1
2
2
2
2
1
2
1
n
n
s
n
n
s
n
s
n
s
df
32. Learning Check
• For an independent-measures research study,
the value of Cohen’s d or r2 helps to describe
______
• the risk of a Type I errorA
• the risk of a Type II errorB
• how much difference there is between the two
treatmentsC
• whether the difference between the two
treatments is likely to have occurred by chanceD
33. Learning Check - Answer
• For an independent-measures research study,
the value of Cohen’s d or r2 helps to describe
______
• the risk of a Type I errorA
• the risk of a Type II errorB
• how much difference there is between the two
treatmentsC
• whether the difference between the two
treatments is likely to have occurred by chanceD
34. Learning Check
• Decide if each of the following statements
is True or False
• The homogeneity assumption requires
the two sample variances to be equalT/F
• If a researcher reports that t(6) = 1.98,
p > .05, then H0 was rejectedT/F
35. Learning Check - Answers
• The assumption requires equal
population variances but test is valid if
sample variances are similar
False
• H0 is rejected when p < .05, and
t > the critical value of tFalse
Instructors may wish to bring to students’ attention the fact that there are many terms in wide use applied to this distinction, so they need to be prepared to learn the fundamentals of the distinction and not just memorize some phrases.
FIGURE 10.1 Do the achievement scores for children taught by method A differ from the scores for children taught by method B? In statistical terms, are the two population means the same or different? Because neither of the two population means is known, it will be necessary to take two samples, one from each population. The first sample provides information about the mean for the first populations, and the second sample provides information about the second population.
Some instructors may want to advise students that the material in this box is designed to be background support for understanding variability and that some students find it very insightful.
FIGURE 10.2 Two population distributions. The scores in population I vary from 50 to 70 (a 20-point spread), and the scores in population II range from 20 to 30 (a 10-point spread). If you select one score from each of these two populations, the closest two values are X1 = 50 and X2 = 30. The two values that are the farthest apart are X1 = 70 and X2 = 20.
FIGURE 10.3 The critical region for the independent-measures hypothesis test in Example 10.1 with df = 18 and α = .01.
FIGURE 10.4 The two groups of scores from Example 10.1 combined into a single distribution. The original scores, including the treatment effect, are shown in part (a). Part (b) shows the adjusted scores, after the treatment effect has been removed.
FIGURE 10.5 The 95% confidence interval for the population mean difference (μ1 - μ2) from Example 10.3. Note that μ1 - μ2 = 0 is excluded from the confidence interval, indicating that a zero difference is not an acceptable value (H0 would be rejected in a hypothesis test with α = .05).
FIGURE 10.6 Two sample distributions representing two different treatments. These data show a significant difference between treatments, t(16) = 8.62, p < .01, and both measures of effect size indicate a large treatment effect, d = 4.10 and r2 = 0.82.
FIGURE 10.7 Two sample distributions representing two different treatments. These data show exactly the same mean difference as the scores in Figure 10.6; however, the variance has been greatly increased. With the increased variance, there is no longer a significant difference between treatments, t(16) = 1.59, p > .05, and both measures of effect size are substantially reduced, d = 0.75 and r2 = 0.14.
FIGURE 10.8 The SPSS output for the independent-measures hypothesis test in Example 10.1.
