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

220 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
2. When computing probabilities for sample means, the most common error is to use standard
deviation (σ) instead of standard error (σ M
) in the z-score formula. Standard deviation
measures the typical deviation (or “error”) for a single score. Standard error measures
the typical deviation (or error) for a sample. Remember: The larger the sample is, the
more accurately the sample represents the population. Thus, sample size (n) is a critical
part of the standard error.
Standard error s M
5 s Ïn
3. Although the distribution of sample means is often normal, it is not always a normal
distribution. Check the criteria to be certain the distribution is normal before you use the
unit normal table to find probabilities (see item 1a of the Summary). Remember that all
probability problems with a normal distribution are easier if you sketch the distribution
and shade in the area of interest.
DEMONSTRATION 7.1
PROBABILITY AND THE DISTRIBUTION OF SAMPLE MEANS
A population forms a normal distribution with a mean of μ = 60 and a standard deviation
of σ = 12. For a sample of n = 36 scores from this population, what is the probability of
obtaining a sample mean greater than 63?
p(M > 63) = ?
STEP 1
Rephrase the probability question as a proportion question. Out of all the possible
sample means for n = 36, what proportion will have values greater than 63? All the possible
sample means is simply the distribution of sample means, which is normal, with a mean of
μ = 60 and a standard error of
s M
5 s Ïn 5 12
Ï36 5 12 6 5 2
The distribution is shown in Figure 7.13.
s M 5 2
FIGURE 7.13
The distribution for
Demonstration 7.1.
60 63
m
0 1 2
M
z