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

SECTION 9.3 | Measuring Effect Size for the t Statistic 281
M 5 13
s 5 3
s 5 3
FIGURE 9.5
The sample distribution for the
scores that were used in Examples
9.2 and 9.3. The population
mean, μ = 10 second, is the
value that would be expected if
attractiveness has no effect on
the infants’ behavior. Note that
the sample mean is displaced
away from μ = 10 by a distance
equal to one standard deviation.
Frequency
3
2
1
8 9 10 11 12 13 14 15 16 17 18
Time spent looking at the attractive face (in seconds)
m 5 10
(from H 0 )
The following example is an opportunity for you to test your understanding of
hypothesis testing and effect size with the t statistic.
EXAMPLE 9.4
A sample of n = 16 individuals is selected from a population with a mean of μ = 40. A
treatment is administered to the individuals in the sample and, after treatment, the sample
has a mean of M = 44 and a variance of s 2 = 16. Use a two-tailed test with α = .05 to
determine whether the treatment effect is significant and compute Cohen’s d to measure the
size of the treatment effect. You should obtain t = 4.00 with df = 15, which is large enough
to reject H 0
with Cohen’s d = 1.00.
■
■ Measuring the Percentage of Variance Explained, r 2
An alternative method for measuring effect size is to determine how much of the variability
in the scores is explained by the treatment effect. The concept behind this measure is that
the treatment causes the scores to increase (or decrease), which means that the treatment is
causing the scores to vary. If we can measure how much of the variability is explained by
the treatment, we will obtain a measure of the size of the treatment effect.
To demonstrate this concept we will use the data from the hypothesis test in Example 9.2.
Recall that the null hypothesis stated that the treatment (the attractiveness of the faces) has
no effect on the infants’ behavior. According to the null hypothesis, the infants should show
no preference between the two photographs, and therefore should spend an average of μ =
10 out of 20 seconds looking at the attractive face.
However, if you look at the data in Figure 9.5, the scores are not centered at μ = 10.
Instead, the scores are shifted to the right so that they are centered around the sample mean,
M = 13. This shift is the treatment effect. To measure the size of the treatment effect we
calculate deviations from the mean and the sum of squared deviations, SS, two different
ways.
Figure 9.6(a) shows the original set of scores. For each score, the deviation from μ = 10
is shown as a colored line. Recall that μ = 10 is from the null hypothesis and represents the
population mean if the treatment has no effect. Note that almost all of the scores are located
on the right-hand side of μ = 10. This shift to the right is the treatment effect. Specifically,
the preference for the attractive face has caused the infants to spend more time looking