Statistics for the Behavioral Sciences by Frederick J. Gravetter, Larry B. Wallnau ISBN 10: 1305504917 ISBN 13: 9781305504912

SECTION 10.4 | Effect Size and Confidence Intervals for the Independent-Measures t 319
EXAMPLE 10.6
In an independent-measures study with n = 16 scores in each treatment, one sample has
M = 89.5 with SS = 1005 and the second sample has M = 82.0 with SS = 1155. The
data produce t(30) = 2.50. Use these data to compute Cohen’s d and r 2 for these data. You
should find that d = 0.883 and r 2 = 0.172.
■
■ Confidence Intervals for Estimating m 1
2 m 2
As noted in chapter 9, it is possible to compute a confidence interval as an alternative method
for measuring and describing the size of the treatment effect. For the single-sample t, we
used a single sample mean, M, to estimate a single population mean. For the independentmeasures
t, we use a sample mean difference, M 1
− M 2
, to estimate the population mean
difference, μ 1
− μ 2
. In this case, the confidence interval literally estimates the size of the
population mean difference between the two populations or treatment conditions.
As with the single-sample t, the first step is to solve the t equation for the unknown
parameter. For the independent-measures t statistic, we obtain
m 1
2m 2
5 M 1
2 M 2
6 ts sM1 2M 2
d (10.10)
In the equation, the values for M 1
− M 2
and for s sM1
are obtained from the sample data.
2M 2
d
Although the value for the t statistic is unknown, we can use the degrees of freedom for the
t statistic and the t distribution table to estimate the t value. Using the estimated t and the
known values from the sample, we can then compute the value of μ 1
− μ 2
. The following
example demonstrates the process of constructing a confidence interval for a population
mean difference.
EXAMPLE 10.7
Earlier we presented a research study comparing puzzle-solving scores for students who
were tested in a dimly lit room scores for students tested in a well-lit room (p. 310). The
results of the hypothesis test indicated a significant mean difference between the two populations
of students. Now, we will construct a 95% confidence interval to estimate the size
of the population mean difference.
The data from the study produced a mean grade of M = 12 for the group in the dimly lit
room and a mean of M = 8 for the group in the well-lit room The estimated standard error for
the mean difference was s sM1
= 1.5. With n = 8 scores in each sample, the independentmeasures
t statistic has df = 14. To have 95% confidence, we simply estimate that the t statis-
2M 2
d
tic for the sample mean difference is located somewhere in the middle 95% of all the possible
t values. According to the t distribution table, with df = 14, 95% of the t values are located
between t = +2.145 and t = –2.145. Using these values in the estimation equation, we obtain
m 1
2m 2
5 M 1
2 M 2
6 ts sM1 2M 2
d
= 12 − 8 ± 2.145(1.5)
= 4 ± 3.218
This produces an interval of values ranging from 4 − 3.218 = 0.782 to 4 + 3.218 = 7.218.
Thus, our conclusion is that students who were tested in the dimly lit room had higher
scores that those who were tested in a well-lit room, and the mean difference between the
two populations is somewhere between 0.782 points and 7.218 points. Furthermore, we are
95% confident that the true mean difference is in this interval because the only value estimated
during the calculations was the t statistic, and we are 95% confident that the t value
is located in the middle 95% of the distribution. Finally, note that the confidence interval
is constructed around the sample mean difference. As a result, the sample mean difference,
M 1
− M 2
= 12 − 8 = 4 points, is located exactly in the center of the interval. ■