Chapter 10—Sampling distributions
MULTIPLE CHOICE
1. As a general rule, the normal distribution is used to approximate the sampling distribution of the
sample mean only if:
A.
the sample size n is sufficiently large.
B.
the sample mean is close to 0.50.
C.
the underlying population is normal.
D.
np and n(1 – p) are both greater than 5.
2. Random samples of size 64 are taken from an infinite population whose mean is 160 and standard
deviation is 32. The mean and standard error of the sample mean, respectively, are:
A.
160 and 32.
B.
64 and 32.
C.
160 and 4.
D.
20 and 4.
3. A normally distributed population with 200 elements has a mean of 60 and a standard deviation of 10.
The probability that the mean of a sample of 25 elements taken from this population will be smaller
than 56 is:
A.
0.0166.
B.
0.0228.
C.
0.3708.
D.
0.0394.
4. Given an infinite population with a mean of 75 and a standard deviation of 12, the probability that the
mean of a sample of 36 observations, taken at random from this population, exceeds 78 is:
A.
0.4332.
B.
0.0668.
C.
0.0987.
D.
0.9013.
5. A population that consists of 250 items has a mean of 37 and a standard deviation of 13. A sample of
size 5 is taken at random from this population. The standard error of the sample mean equals (up to
three decimal places):
A.
1.838.
B.
1.648.
C.
0.822.
D.
2.715.
6. An infinite population has a mean of 33 and a standard deviation of 6. A sample of 100 observations is
to be taken at random from this population. The probability that the sample mean will be between 34.5
and 36.1 is:
A.
0.1543.
B.
0.2960.
C.
0.6046.
D.
0.4503.
7. If all possible samples of size n are drawn from an infinite population with a mean of 15 and a
standard deviation of 5, then the standard error of the sample mean equals 1.0 only for samples of size:
A.
5.
B.
15.
C.
25.
D.
75.
8. As a general rule in computing the standard error of the sample mean, the finite population correction
factor is used only if the:
A.
sample size is smaller than 10% of the population size.
B.
population size is smaller than 10% of the sample size.
C.
sample size is greater than 1% of the population size.
D.
population size is greater than 1% of the sample size.
9. Consider an infinite population with a mean of 160 and a standard deviation of 25. A random sample
of size 64 is taken from this population. The standard deviation of the sample mean equals:
A.
12.649.
B.
25.0.
C.
2.56.
D.
3.125.
10. A sample of size 50 is to be taken from an infinite population whose mean and standard deviation are
52 and 20, respectively. The probability that the sample mean will be larger than 49 is:
A.
0.4452.
B.
0.9452.
C.
0.8554.
D.
0.3554.
11. A sample of size n is selected at random from an infinite population. As n increases, which of the
following statements is true?
A.
The population standard deviation decreases.
B.
The standard error of the sample mean decreases.
C.
The population standard deviation increases.
D.
The standard error of the sample mean increases.
12. The finite population correction factor should not be used when:
A.
we are sampling from an infinite population.
B.
we are sampling from a finite population.
C.
sample size is greater than 1% of the population size.
D.
None of the above statements is correct.
13. Random samples of size 81 are taken from an infinite population whose mean and standard deviation
are 45 and 9, respectively. The mean and standard error of the sampling distribution of the sample
mean are:
A.
9 and 45.
B.
45 and 9.
C.
81 and 45.
D.
45 and 1.
14. A sample of size 35 is selected at random from a finite population. If the finite population correction
factor is 0.5, then (rounded to the nearest integer) the population size is:
A.
46.
B.
2116.
C.
78.
D.
35.
15. The central limit theorem states that if a random sample of size n is drawn from a population, then the
sampling distribution of the sample mean
X
:
A.
is approximately normal if n > 30.
B.
is approximately normal if n < 30.
C.
is approximately normal if the underlying population is normal.
D.
has the same variance as the population.
16. The expected value of the sampling distribution of the sample mean
X
equals the population mean :
A.
when the population is normally distributed.
B.
when the population is symmetric.
C.
when the population size N > 30.
D.
for all populations.
17. If all possible samples of size n are drawn from an infinite population with mean and standard
deviation , then the standard error of the sample mean is inversely proportional to:
A.
B.
C.
n.
D.
n
.
18. The standard deviation of the sampling distribution of the sample mean is also called the:
A.
central limit theorem.
B.
standard error of the mean.
C.
finite population correction factor.
D.
population standard deviation.
19. If a random sample of size n is drawn from a normal population, then the sampling distribution of the
sample mean
X
will be:
A.
normal for all values of n.
B.
normal only for n > 30.
C.
approximately normal for all values of n.
D.
approximately normal only for n > 30.
20. If all possible samples of size n are drawn from a population, the probability distribution of the sample
mean
X
is called the:
A.
standard error of
X
.
B.
expected value of
X
.
C.
sampling distribution of
X
.
D.
normal distribution.
TRUE/FALSE
1. The central limit theorem is basic to the concept of statistical inference, because it permits us to draw
conclusions about the population based strictly on sample data, and without having any knowledge
about the distribution of the underlying population.
2. When a great many simple random samples of size n are drawn from a population that is normally
distributed, the sampling distribution of the sample means will be normal, regardless of sample size n.
3. The standard error of the mean is the standard deviation of the sampling distribution of
X
.
4. The standard deviation of a sampled population is also called the standard error of the sample mean.
5. Consider an infinite population with a mean of 100 and a standard deviation of 20. A random sample
of size 50 is taken from this population. The standard deviation of the sample mean equals 3.2.
6. If all possible samples of size n are drawn from an infinite population with a mean of 60 and a
standard deviation of 8, then the standard error of the sample mean equals 1.0 only for samples of size
64.
7. If all possible samples of size n are drawn from a population, the probability distribution of the sample
mean
X
is referred to as the normal distribution.
8. A sample of size n is selected at random from an infinite population. As n increases the standard error
of the sample mean decreases.
9. A sample of size 70 is selected at random from a finite population. If the finite population correction
factor is 0.808, the population size (rounded to the nearest integer) must be 200.
SHORT ANSWER
1. A researcher conducted a survey on a university campus for a sample of 64 third-year students and
reported that third-year students read an average of 3.12 books in the prior academic semester, with a
standard deviation of 2.15 books. Determine the probability that the sample mean is:
a. above 3.45.
b. between 3.38 and 3.58.
c. below 2.94.
2. An infinite population has a mean of 120 and a standard deviation of 44. A sample of 100 observations
is to be selected at random from the population.
a. What is the expected value of the sample mean?
b. What is the standard deviation of the sample mean?
c. What is the shape of the sampling distribution of the sample mean?
d. What does the sampling distribution of the sample mean show?
3. If the weekly demand for packs of soft drink at a supermarket is normally distributed with a mean of
47.6 packs and a standard deviation of 5.8 packs, what is the probability that the average demand for a
sample of 10 supermarkets will exceed 50 packs in a given week?
4. Suppose that the time needed to complete a final exam is normally distributed with a mean of 85
minutes and a standard deviation of 18 minutes.
a. What is the probability that the total time taken by a group of 100 students will not exceed 8200
minutes?
b. What assumption did you have to make in your computations in part (a)?
5. The heights of 9-year-old children are normally distributed, with a mean of 123 cm and a standard
deviation of 10 cm.
a. Find the probability that one randomly selected 9-year-old child is taller than 125 cm.
b. Find the probability that three randomly selected 9-year-old children are both taller than 125 cm.
c. Find the probability that the mean height of three randomly selected 9-year-old children is greater
than 125 cm.
6. Find the sampling distribution of the sample mean
X
if samples of size 2 are drawn from the
following population.
x
1
2
3
p(x)
0.2
0.3
0.5
7. An infinite population has a mean of 100 and a standard deviation of 20. Suppose that the population is
not normally distributed. What does the central limit theorem say about the sampling distribution of
the mean if samples of size 64 are drawn at random from this population?
8. Suppose that the average annual income of a lawyer is $150 000 with a standard deviation of $40 000.
Assume that the income distribution is normal.
a. What is the probability that the average annual income of a sample of 5 lawyers is more than
$120 000?
b. What is the probability that the average annual income of a sample of 15 lawyers is more than
$120 000?
9. Assume that the time needed by a worker to perform a maintenance operation is normally distributed,
with a mean of 60 minutes and a standard deviation of 6 minutes. What is the probability that the
average time needed by a sample of 5 workers to perform the maintenance is between 63 and 68
minutes?
10. In order to estimate the mean salary for a population of 500 employees, the managing director of a
certain company selected at random a sample of 40 employees.
a. Would you use the finite population correction factor in calculating the standard error of the
sample mean? Explain.
b. If the population standard deviation is $800, compute the standard error both with and without
using the finite population correction factor.
c. What is the probability that the sample mean salary of the employees will be within ±$200 of the
population mean salary?
11. A sample of 50 observations is drawn at random from a normal population whose mean and standard
deviation are 75 and 6, respectively.
a. What does the central limit theorem say about the sampling distribution of the sample mean?
Why?
b. Find the mean and standard error of the sampling distribution of the sample mean.
c. Find P(
X
> 73).
d. Find P(
X
< 74).
12. Find the sampling distribution of the sample mean
X
if samples of size 2 are drawn from the
following population.
x
3
5
p(x)
0.6
0.4
13. A sample of size 400 is drawn from a population whose mean and variance are 5000 and 10 000,
respectively. Find the following probabilities.
a. P(
X
< 4,990).
b. P(4995 <
X
< 5010).
c. P(
X
= 5000).
14. A sample of 30 observations is drawn from a normal population with mean of 750 and a standard
deviation of 300. Suppose the population size is 600.
a. Find the expected value of the sample mean
X
.
b. Find the standard error of the sample mean
X
.
c. Find P(
X
> 790).
d. Find P(
X
< 650).
e. Find P(760 <
X
< 810).
X
15. An auditor knows from past history that the average accounts receivable for a company is $521.72,
with a standard deviation of $584.64. If the auditor takes a simple random sample of 100 accounts,
what is the probability that the mean of the sample will be within $120 of the population mean?
16. In a given year, the average annual salary of an AFL player was $205 000, with a standard deviation of
$24 500. If a simple random sample of 50 players is taken, what is the probability that the sample
mean will exceed $210 000?
17. The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a
standard deviation of 10 centimetres. Use this information to answer the following question(s).
What is the probability that a randomly selected woman is shorter than 162 centimetres?
18. The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a
standard deviation of 10 centimetres. Use this information to answer the following question(s).
A random sample of five women is selected. What is the probability that the sample mean is smaller
than 162 centimetres?
19. The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a
standard deviation of 10 centimetres. Use this information to answer the following question(s).
What is the probability that the mean height of a random sample of 30 women is smaller than 162
centimetres?
20. The heights of women in Australia are normally distributed, with a mean of 165 centimetres and a
standard deviation of 10 centimetres. Use this information to answer the following question(s).
If the population of women’s heights were not normally distributed, which, if any, of the following
questions could you answer?
a. What is the probability that a randomly selected woman is taller than 160 cm?
b. A random sample of five women is selected. What is the probability that the sample mean is
greater than 160 cm?
c. What is the probability that the mean height of a random sample of 75 women is greater than 160
cm?
21. The amount of time spent by Australian adults playing sports per day is normally distributed, with a
mean of 4 hours and standard deviation of 1.25 hours. Use this information to answer the following
question(s).
Find the probability that a randomly selected Australian adult plays sport for more than 5 hours per
day.
22. The amount of time spent by Australian adults playing sports per day is normally distributed, with a
mean of 4 hours and standard deviation of 1.25 hours. Use this information to answer the following
question(s).
Find the probability that if four Australian adults are randomly selected, their average number of hours
spent playing sport is more than 5 hours per day.
23. The amount of time spent by Australian adults playing sports per day is normally distributed, with a
mean of 4 hours and standard deviation of 1.25 hours. Use this information to answer the following
question(s).
Find the probability that if four Australian adults are randomly selected, all four play sport for more
than 5 hours per day.
24. The following table gives the number of pets owned for a population of four families.
Family
A
B
C
D
Number of pets owned
2
1
4
3
Find the mean and standard deviation for the population.
25. The following table gives the number of pets owned for a population of four families.
Family
A
B
C
D
Number of pets owned
2
1
4
3
Samples of size 2 will be drawn at random from the population. Use your answers to the previous
question to calculate the mean and the standard deviation of the sampling distribution of the sample
means.
26. The following table gives the number of pets owned for a population of four families.
Family
A
B
C
D
Number of pets owned
2
1
4
3
List all possible samples of two families that can be selected without replacement from this population,
and compute the sample mean for each sample.
27. The following table gives the number of pets owned for a population of four families.
Family
A
B
C
D
Number of pets owned
2
1
4
3
Find the sampling distribution of
X
.
ANS:
28. Using the following sampling distribution, directly recalculate the mean and standard deviation of
X
.
x
1.5
2.0
2.5
3.0
3.5
p(
x
)
1/6
1/6
1/6
1/6
1/6
ö
p
29. A video rental store wants to know what proportion of its customers are under 21 years old. A simple
random sample of 500 customers is taken, and 375 of them are under 21. Assume that the true
population proportion of customers aged under 21 is 0.68.
Describe the sampling distribution of proportion of customers who are under age 21.
30. A video rental store wants to know what proportion of its customers are under 21 years old. A simple
random sample of 500 customers is taken, and 375 of them are under 21. Assume that the true
population proportion of customers aged under 21 is 0.68.
Find the mean and standard deviation of
ö
p
.
31. A video rental store wants to know what proportion of its customers are under 21 years old. A simple
random sample of 500 customers is taken, and 375 of them are under 21. Assume that the true
population proportion of customers aged under 21 is 0.68.
What is the probability that the sample proportion
ö
p
will be within 0.03 of the true proportion of
customers who are aged under 21?
x
1.5
2.0
2.5
3.0
3.5
p(
x
)
1/6
1/6
1/6
1/6
1/6
32. A local newspaper sells an average of 2100 papers per day, with a standard deviation of 500 papers.
Consider a sample of 60 days of operation.
a. What is the shape of the sampling distribution of the sample mean number of papers sold per day?
Why?
b. Find the expected value and the standard error of the sample mean.
c. What is the probability that the sample mean will be between 2000 and 2300 papers?