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

206 CHAPTER 7 | Probability and Samples: The Distribution of Sample Means
LEARNING CHECK
1. If samples are selected from a population with μ = 80 and σ = 12, then which of
the following samples will have the largest expected value for M?
a. n = 10 scores with M = 82
b. n = 20 scores with M = 84
c. n = 30 scores with M = 86
d. All of the samples will have the same expected value.
2. If random samples, each with n = 4 scores are selected from a population with
µ = 80 and σ = 12, then how much distance is expected on average between the
sample means and the population mean?
a. 4(12) = 48 points
b. 12 points
c. 12
4 = 3 points
12
d. = 6 points
Ï4
3. The standard distance between a sample mean and the population mean is 6 points
for samples of n = 16 scores selected from a population with a mean of μ = 50.
What is the standard deviation for the population?
a. 48
b. 24
c. 6
d. 3
ANSWERS
1. D, 2. D, 3. B
7.3 Probability and the Distribution of Sample Means
LEARNING OBJECTIVES
4. Calculate the z-score for a sample mean.
5. Describe the circumstances in which the distribution of sample means is normal and,
in these circumstances, find the probability associated with a specific sample.
The primary use of the distribution of sample means is to find the probability associated
with any specific sample. Recall that probability is equivalent to proportion. Because the
distribution of sample means presents the entire set of all possible sample means, we can
use proportions of this distribution to determine probabilities. The following example demonstrates
this process.
EXAMPLE 7.3
The population of scores on the SAT forms a normal distribution with μ = 500 and
σ = 100. If you take a random sample of n = 16 students, what is the probability that the
sample mean will be greater than M = 525?
First, you can restate this probability question as a proportion question: Out of all the
possible sample means, what proportion has values greater than 525? You know about “all
the possible sample means”; this is the distribution of sample means. The problem is to find
a specific portion of this distribution.