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19. The term 1 – refers to the level of confidence that a confidence interval does not contain the population
parameter.
20. In the formula
/ 2 /nx z
, the subscript
/ 2
refers to the area in the lower tail or upper tail of the
sampling distribution of the sample mean.
21. The larger the level of confidence used in constructing a confidence interval estimate of the population
mean, the narrower the confidence interval.
22. It is impossible to construct a confidence interval estimate of the population mean if the population
variance is unknown.
23. If there are two unbiased estimators of a parameter, the one whose variance is smaller is said to be
relatively efficient.
24. In developing an interval estimate for a population mean, the population standard deviation
was
assumed to be 8. The interval estimate was 50.0 ± 2.50. Had
equalled 16, the interval estimate
would have been 100 ± 5.0.
25. The sample mean is an unbiased estimator of the population mean
, and (when sampling from a
normal population) the sample median is also an unbiased estimator of
. However, the sample
median is relatively more efficient than the sample mean.
26. A 95% confidence interval estimate for a population mean
is determined to be 75 to 85. If the
confidence level is reduced to 80%, the confidence interval for
becomes narrower.
27. When constructing confidence interval for a parameter, we generally set the confidence level
1
−
close to 1 (usually between 0.90 and 0.99) because we would like to be reasonably confident that the
interval includes the actual value of the population parameter.
28. Suppose that a 90% confidence interval for
is given by
0.75x
. This notation means that we are
90% confident that
falls between
0.75x−
and
0.75x+
.
29. We cannot interpret the confidence interval estimate of
as a probability statement about
, simply
because the population mean is a fixed but unknown quantity.
30. The width of the confidence interval estimate of the population mean
is a function of only two
quantities, the population standard deviation
and the sample size n.
31. At a given sample size and level of confidence, the smaller the population standard deviation
, the
wider and thus the less precise the confidence interval estimate of
is.
32. In general, increasing the confidence level
1
−
will narrow the interval, and decreasing it widens the
interval.
33. Suppose that a 95% confidence interval for
is given by
3.25x
. This notation means that if we
repeatedly draw samples of the same size from the same population, 95% of the values of
x
will be
such that
would lie somewhere between
3.25x−
and
3.25x+
.
34. When constructing a confidence interval estimate of
, doubling the sample size n reduces the width
%of the interval by half.
35. You need four values to construct the confidence interval estimate of
. These are the sample mean,
sample size, population standard deviation and confidence level.
36. In an effort to identify the true proportion of first-year university students who are under 18 years of
age, a random sample of 500 first-year students was taken. Only 50 of them were under the age of 18.
The value 0.10 would be used as a point estimate to the true proportion of first-year students aged
under 18.
37. The mean of the sampling distribution of the sample proportion
ˆ
p
, when the sample size n = 100 and
the population proportion p = 0.92, is 92.0.
38. The standard error of the sampling distribution of the sample proportion
ˆ
p
, when the sample size n =
100 and the population proportion p = 0.30, is 0.0021.
39. Recall the rule of thumb used to indicate when the normal distribution is a good approximation to the
sampling distribution for the sample proportion
ˆ
p
. For the combination n = 50; p = 0.05, the rule is
satisfied.
40. As a general rule, the normal distribution is used to approximate the sampling distribution of the
sample proportion only if both n
ˆ
p
and n
ˆ
q
are at least five (
ˆ
q
= 1 –
ˆ
p
).
SHORT ANSWER
1. A survey of 100 retailers revealed that the mean after-tax profit was $80 000. Assuming that the
population standard deviation is $15 000, determine the 95% confidence interval estimate of the mean
after-tax profit for all retailers.
2. The nighttime temperature readings for 20 winter days in Sydney are normally distributed with a mean
of 5.5ºC and a population standard deviation of 1.5ºC. Determine the 90% confidence interval estimate
for the mean winter nighttime temperature.
3. A sample of 50 students was asked how much time they spend on average a week in front of a
computer. The sample mean and sample standard deviation were 15.8 and 2.7 hours, respectively.
Estimate with 95% confidence interval the mean number of hours students spend in front of a
computer a week.
4. A random sample of 10 waitresses in Darwin revealed the following hourly earnings (in dollars,
including tips):
19
18
15
16
18
17
16
18
20
14
If the hourly earnings are normally distributed with a standard deviation of $4.5, estimate with 95%
confidence the mean hourly earnings for all waitresses in Darwin.
5. Determine the sample size that is required to estimate a population mean to within 0.4 units with a
99% confidence when the population standard deviation is 1.75.
6. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 95% confidence interval estimate of the population mean.
7. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Interpret what the confidence interval estimate tells you.
8. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 99% confidence interval estimate of the population mean.
9. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 90% confidence interval estimate of the population mean.
10. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 95% confidence interval estimate of the population mean, changing the sample size to
300.
11. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 95% confidence interval estimate of the population mean, changing the sample size to
36.
12. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 95% confidence interval estimate of the population mean, changing the population
standard deviation to 2.
13. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 95% confidence interval estimate of the population mean, changing the population
standard deviation to 1.2.
14. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 95% confidence interval estimate of the population mean, changing the sample mean to
5.0 hours.
15. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
Determine the 95% confidence interval estimate of the population mean, changing the sample mean to
8.5 hours.
16. Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a
standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean
is computed as 6.5 hours.
What happens to the width of the confidence interval estimate when each of the following things
happens?
a. The confidence level increases.
b. The confidence level decreases.
c. The sample size increases.
d. The sample size decreases.
e. The value of the population standard deviation increases.
f. The value of the population standard deviation decreases.
g. The value of the sample mean increases.
h. The value of the sample mean decreases.
17. Suppose that your task is to estimate the mean of a normally distributed population to within 10 units
with 95% confidence and that the population standard deviation is known to be 70.
What sample size should you use?
18. Suppose that your task is to estimate the mean of a normally distributed population to within 10 units
with 95% confidence and that the population standard deviation is known to be 70.
What sample size should you use, changing the standard deviation to 50?
19. Suppose that your task is to estimate the mean of a normally distributed population to within 10 units
with 95% confidence and that the population standard deviation is known to be 70.
What sample size should you use, changing the standard deviation to 90?
20. Suppose that your task is to estimate the mean of a normally distributed population to within 10 units
with 95% confidence and that the population standard deviation is known to be 70.
What sample size should you use with a 90% confidence level?
21. Suppose that your task is to estimate the mean of a normally distributed population to within 10 units
with 95% confidence and that the population standard deviation is known to be 70.
What sample size should you use with a 99% confidence level?
22. Suppose that your task is to estimate the mean of a normally distributed population to within 10 units
with 95% confidence and that the population standard deviation is known to be 70.
What sample size should be used if you wish to estimate the population mean to within 20 units?
23. Suppose that your task is to estimate the mean of a normally distributed population to within 10 units
with 95% confidence and that the population standard deviation is known to be 70.
What sample size should be used if you wish to estimate the population mean to within 5 units?
24. A financial analyst wants to determine the mean annual return on mutual funds. A random sample of
60 returns shows a mean of 12%. If the population standard deviation is assumed to be 4%, estimate
with 95% confidence the mean annual return on all mutual funds.
25. A normal population has a standard deviation of 15. How large a sample should be drawn to estimate
the population mean to within 1.5 with 95% confidence?
26. How large a sample of state employees should be taken if we want to estimate with 98% confidence
the mean salary to within $2000. The population standard deviation is assumed to be $10 500.
27. A random sample of 10 university students was surveyed to determine the amount of time they spent
weekly using a personal computer. The times (in hours) were:
13
14
5
6
8
10
7
12
15
3
If the times are normally distributed with a standard deviation of 5.2 hours, estimate with 90%
confidence the mean weekly time spent using a personal computer for all university students.
28. A statistician wants to estimate the mean weekly family expenditure on clothes. He believes that the
largest weekly expenditure is $650 and the lowest is $150.
a. Estimate the standard deviation of the weekly expenditure.
b. Determine with 99% confidence the number of families that must be sampled to estimate the mean
weekly family expenditure on clothes to within $15.
29. How large a sample must be drawn to estimate the proportion of students who prefer statistics over
mathematics, to within 0.02 with 95% confidence?
30. How large a sample should be taken to estimate a population proportion to within 0.01 with 90%
confidence if the proportion is known to be around 5%?
31. How large a sample must be drawn to estimate a population proportion to within 0.03 with 95%
confidence if we believe that the proportion lies somewhere between 25% and 45%?
32. As a manufacturer of golf clubs, a major corporation wants to estimate the proportion of golfers who
are right-handed. How many golfers must be surveyed if they want to be within 0.02 with a 95%
confidence level:
a. assuming that there is no information available that could be used as an estimate of p?
b. assuming that the manufacturer has an estimate of p obtained from a previous study that suggests
that 75% of golfers are right-handed?
33. A marketing researcher wishes to determine the sample size needed to estimate the proportion of wine
drinkers who prefer a certain brand of wine. How many wine drinkers should be surveyed if the
researcher wants to be within 0.025 with 95% confidence?
34. The probability of a success on any trial of a binomial experiment is 0.15. Find the probability that the
proportion of success in a sample of 300 is more than 12%.
35.A politician believes that the proportion of voters who will vote for a Coalition candidate in the 2004
general election is 0.65. A sample of 500 voters is selected at random.
a. Assume that the politician is correct and p = 0.65. What is the sampling distribution of the sample
proportion
ˆ
p
? Explain.
b. Find the expected value and standard deviation of the sample proportion
ˆ
p
.
c. What is the probability that the number of voters in the sample who will vote for a Labor candidate
in the 2004 general election will be between 340 and 350?
36. A simple random sample of 300 observations is taken from a population. Assume that the population
proportion p = 0.6.
a. What is the expected value of the sample proportion
ˆ
p
?
b. What is the standard error of the sample proportion
ˆ
p
?
c. What is the probability that the sample proportion
ˆ
p
will be within ±0.02 of the population
proportion p?
37. The head of the statistics department in a certain university believes that 70% of the department’s
graduate assistantships are given to international students. A random sample of 50 graduate assistants
is taken.
a. Assume that the chairman is correct and p = 0.70. What is the sampling distribution of the sample
proportion
ˆ
p
Explain.
b. Find the expected value and the standard error of the sampling distribution of
ˆ
p
.
c. What is the probability that the sample proportion
ˆ
p
will be between 0.65 and 0.73?
d. What is the probability that the sample proportion
ˆ
p
will be within ±0.05 of the population
proportion p?
