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

SECTION 7.3 | Probability and the Distribution of Sample Means 207
Caution: Whenever
you have a probability
question about a sample
mean, you must use the
distribution of sample
means.
Although we cannot construct the distribution of sample means by repeatedly taking
samples and calculating means (as in Example 7.1), we know exactly what the distribution
looks like based on the information from the central limit theorem. Specifically, the distribution
of sample means has the following characteristics:
1. The distribution is normal because the population of SAT scores is normal.
2. The distribution has a mean of 500 because the population mean is μ = 500.
3. For n = 16, the distribution has a standard error of σ M
= 25:
s M
5 s Ïn 5 100
Ï16 5 100
4 5 25
This distribution of sample means is shown in Figure 7.5.
We are interested in sample means greater than 525 (the shaded area in Figure 7.5), so
the next step is to use a z-score to locate the exact position of M = 525 in the distribution.
The value 525 is located above the mean by 25 points, which is exactly 1 standard deviation
(in this case, exactly 1 standard error). Thus, the z-score for M = 525 is z = +1.00.
Because this distribution of sample means is normal, you can use the unit normal table
to find the probability associated with z = +1.00. The table indicates that 0.1587 of the
distribution is located in the tail of the distribution beyond z = +1.00. Our conclusion is
that it is relatively unlikely, p = 0.1587 (15.87%), to obtain a random sample of n = 16
students with an average SAT score greater than 525.
■
■ A z-Score for Sample Means
As demonstrated in Example 7.3, it is possible to use a z-score to describe the exact location
of any specific sample mean within the distribution of sample means. The z-score tells
exactly where the sample mean is located in relation to all the other possible sample means
that could have been obtained. As defined in Chapter 5, a z-score identifies the location
with a signed number so that
1. The sign tells whether the location is above (+) or below (–) the mean.
2. The number tells the distance between the location and the mean in terms of the
number of standard deviations.
s M 525
FIGURE 7.5
The distribution of sample
means for n = 16. Samples
were selected from a normal
population with μ = 500 and
σ = 100.
500 525
m
550
0 1 2
M
z