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

PROBLEMS 221
STEP 2
Compute the z-score for the sample mean. A sample mean of M = 63 corresponds to a
z-score of
z 5 M 2m 63 2 60
5 5 3 s M
2 2 5 1.50
Therefore, p(M > 63) = p(z > 1.50)
STEP 3
Look up the proportion in the unit normal table. Find z = 1.50 in column A and read
across the row to find p = 0.0668 in column C. This is the answer.
p(M > 63) = p(z > 1.50) = 0.0668 (or 6.68%)
PROBLEMS
1. Briefly define each of the following:
a. Distribution of sample means
b. Expected value of M
c. Standard error of M
2. A sample is selected from a population with a mean of
μ = 40 and a standard deviation of σ = 8.
a. If the sample has n = 4 scores, what is the
expected value of M and the standard error of M?
b. If the sample has n = 16 scores, what is the
expected value of M and the standard error of M?
3. Describe the distribution of sample means (shape,
mean, standard error) for samples of n = 64 selected
from a population with a mean of μ = 90 and a standard
deviation of σ = 32.
4. The distribution of sample means is not always a
normal distribution. Under what circumstances is the
distribution of sample means not be normal?
5. A population has a standard deviation of σ = 24.
a. On average, how much difference should there be
between the sample mean and the population mean
for a random sample of n = 4 scores from this
population?
b. On average, how much difference should there be
for a sample of n = 9 scores?
c. On average, how much difference should there be
for a sample of n = 16 scores?
6. For a population with a mean of μ = 45 and a standard
deviation of σ = 10, what is the standard error
of the distribution of sample means for each of the
following sample sizes?
a. n = 4 scores
b. n = 25 scores
7. For a population with σ = 12, how large a sample is
necessary to have a standard error that is:
a. less than 4 points?
b. less than 3 points?
c. less than 2 points?
8. If the population standard deviation is σ = 10, how
large a sample is necessary to have a standard error
that is
a. less than 5 points?
b. less than 2 points?
c. less than 1 point?
9. For a sample of n = 16 scores, what is the value
of the population standard deviation (σ) necessary
to produce each of the following a standard error
values?
a. σ M
= 8 points?
b. σ M
= 4 points?
c. σ M
= 1 point?
10. For a population with a mean of μ = 40 and a standard
deviation of σ = 8, find the z-score corresponding
to each of the following samples.
a. X = 36 for a sample of n = 1 score
b. M = 36 for a sample of n = 4 scores
c. M = 36 for a sample of n = 16 scores
11. A sample of n = 25 scores has a mean of M = 68.
Find the z-score for this sample:
a. If it was obtained from a population with μ = 60
and σ = 10.
b. If it was obtained from a population with μ = 60
and σ = 20.
c. If it was obtained from a population with μ = 60
and σ = 40.
12. A population forms a normal distribution with a mean
of μ = 55 and a standard deviation of σ = 12. For
each of the following samples, compute the z-score for
the sample mean.
a. M = 58 for n = 4 scores
b. M = 58 for n = 16 scores
c. M = 58 for n = 36 scores
13. Scores on a standardized reading test for 4 th -grade
students form a normal distribution with μ = 60 and