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

216 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
Population
of weights
for adult rats
Normal
m 5 400
s 5 20
FIGURE 7.11
The structure of the research
study described in Example 7.7.
The purpose of the study is to
determine whether the treatment
(a growth hormone) has an
effect on weight for rats.
Sample of
n 5 25 rats
T
r
e
a
t
m
e
n
t
Treated
sample of
n 5 25 rats
The distribution of sample means and the standard error can help researchers make this
decision. In particular, the distribution of sample means can be used to show exactly what
would be expected for a sample of rats that do not receive any hormone injections. This
allows researchers to make a simple comparison between
a. The sample of treated rats (from the research study) asdsd
b. Samples of untreated rats (from the distribution of sample means)
If our treated sample is noticeably different from the untreated samples, then we have evidence
that the treatment has an effect. On the other hand, if our treated sample still looks
like one of the untreated samples, then we must conclude that the treatment does not appear
to have any effect.
We begin with the original population of untreated rats and consider the distribution of
sample means for all the possible samples of n = 25 rats. The distribution of sample means
has the following characteristics:
1. It is a normal distribution, because the population of rat weights is normal.
2. It has an expected value of 400, because the population mean for untreated rats is
μ = 400.
3. It has a standard error of σ M
= 20 = 20
Ï25 5 = 4, because the population standard
deviation is σ = 20 and the sample size is n = 25.
The distribution of sample means is shown in Figure 7.12. Notice that a sample of
n = 25 untreated rats (without the hormone) should have a mean weight around 400 grams.
To be more precise, we can use z-scores to determine the middle 95% of all the possible
sample means. As demonstrated in Chapter 6 (page 185), the middle 95% of a normal
distribution is located between z-score boundaries of z = +1.96 and z = –1.96 (check the
unit normal table). These z-score boundaries are shown in Figure 7.12. With a standard
error of σ M
= 4 points, a z-score of z = 1.96 corresponds to a distance of 1.96(4) = 7.84
points from the mean. Thus, the z-score boundaries of ±1.96 correspond to sample means
of 392.16 and 407.84.