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

SECTION 9.2 | Hypothesis Tests with the t Statistic 275
As always, the null hypothesis states that the treatment has no effect; specifically, H 0
states that the population mean is unchanged. Thus, the null hypothesis provides a specific
value for the unknown population mean. The sample data provide a value for the sample
mean. Finally, the variance and estimated standard error are computed from the sample
data. When these values are used in the t formula, the result becomes
t =
sample mean
(from the data)
population mean
– (hypothesized from H0 )
estimated standard error
(computed from the sample data)
As with the z-score formula, the t statistic forms a ratio. The numerator measures the
actual difference between the sample data (M) and the population hypothesis (μ).
The estimated standard error in the denominator measures how much difference is reasonable
to expect between a sample mean and the population mean. When the obtained
difference between the data and the hypothesis (numerator) is much greater than expected
(denominator), we obtain a large value for t (either large positive or large negative). In this
case, we conclude that the data are not consistent with the hypothesis, and our decision
is to “reject H 0
.” On the other hand, when the difference between the data and the hypothesis
is small relative to the standard error, we obtain a t statistic near zero, and our decision
is “fail to reject H 0
.”
The Unknown Population As mentioned earlier, the hypothesis test often concerns
a population that has received a treatment. This situation is shown in Figure 9.3. Note
that the value of the mean is known for the population before treatment. The question is
whether the treatment influences the scores and causes the mean to change. In this case,
the unknown population is the one that exists after the treatment is administered, and the
null hypothesis simply states that the value of the mean is not changed by the treatment.
Although the t statistic can be used in the “before and after” type of research shown
in Figure 9.3, it also permits hypothesis testing in situations for which you do not have
a known population mean to serve as a standard. Specifically, the t test does not require
any prior knowledge about the population mean or the population variance. All you need
to compute a t statistic is a null hypothesis and a sample from the unknown population.
Thus, a t test can be used in situations for which the null hypothesis is obtained from
FIGURE 9.3
The basic research situation
for the t statistic hypothesis
test. It is assumed that the
parameter μ is known for the
population before treatment.
The purpose of the research
study is to determine whether
the treatment has an effect.
Note that the population after
treatment has unknown values
for the mean and the variance.
We will use a sample to test a
hypothesis about the population
mean.
Known population
before treatment
Unknown population
after treatment
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