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

SUMMARY 291
3. A researcher rejects the null hypothesis with a regular two-tailed test using α = .05.
If the researcher used a directional (one-tailed) test with the same data, then what
decision would be made?
a. Definitely reject the null hypothesis.
b. Definitely reject the null hypothesis if the treatment effect is in the predicted
direction.
c. Possibly not reject the null hypothesis even if the treatment effect is in the predicted
direction.
d. It is impossible to determine without more information.
ANSWERS
1. C, 2. B, 3. B
SUMMARY
1. The t statistic is used instead of a z-score for hypothesis
testing when the population standard deviation (or
variance) is unknown.
2. To compute the t statistic, you must first calculate the
sample variance (or standard deviation) as a substitute
for the unknown population value.
sample variance 5 s 2 5 SS
df
Next, the standard error is estimated by substituting s 2
in the formula for standard error. The estimated standard
error is calculated in the following manner:
estimated standard error 5 s M
5Î
s2
n
Finally, a t statistic is computed using the estimated
standard error. The t statistic is used as a substitute for
a z-score that cannot be computed when the population
variance or standard deviation is unknown.
t 5 M 2m
s M
3. The structure of the t formula is similar to that of the
z-score in that
sample mean 2 population mean
z or t 5
sestimatedd standard error
For a hypothesis test, you hypothesize a value for the
unknown population mean and plug the hypothesized
value into the equation along with the sample mean and
the estimated standard error, which are computed from
the sample data. If the hypothesized mean produces an
extreme value for t, you conclude that the hypothesis
was wrong.
4. The t distribution is an approximation of the normal
z distribution. To evaluate a t statistic that is obtained
for a sample mean, the critical region must be located
in a t distribution. There is a family of t distributions,
with the exact shape of a particular distribution of
t values depending on degrees of freedom (n – 1).
Therefore, the critical t values depend on the value
for df associated with the t test. As df increases,
the shape of the t distribution approaches a normal
distribution.
5. When a t statistic is used for a hypothesis test,
Cohen’s d can be computed to measure effect size. In
this situation, the sample standard deviation is used in
the formula to obtain an estimated value for d:
estimated d 5
mean difference
standard deviation 5 M 2m
s
6. A second measure of effect size is r 2 , which measures
the percentage of the variability that is accounted for
by the treatment effect. This value is computed as
follows:
t2
r 2 5
t 2 1 df