Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
When estimating a population mean by a sample mean, which of the following does the margin of
error not depend on?
1)
A)
The sample mean
B)
The confidence level
C)
The sample size
D)
The population standard deviation
2)
In stating a confidence–interval estimate of a population mean, the level of confidence increases as
the size of the interval .
2)
A)
decreases
B)
increases
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
3)
Suppose you wish to use the z–interval procedure to obtain a confidence interval for the
population mean. If the sample size is large, but the data contain outliers, is it reasonable to
use the z–interval procedure? How should you proceed?
3)
4)
Scores on a test are normally distributed with a mean of 71 and a standard deviation of 11.
For samples of size 35, which of the two variables below has more variation? Why? What is
the distribution of each of the variables?
(a) x– 71
11/ 35 (b) x– 71
s/ 35
4)
1
5)
Mary wishes to estimate the mean height of women aged 18–24. She picks a sample of 100
women aged between 18 and 24 and constructs a 99% confidence interval for the
population mean. If she were to repeat this procedure 200 times in total, she would obtain
200 different confidence intervals. How many of these intervals would you expect to
contain the population mean, µ? Explain your thinking.
5)
6)
Compare the basic properties of t–curves with the basic properties of the standard normal
curve. In what ways are t–curves similar to the standard normal curve? In what ways are
they different?
6)
7)
A researcher estimates that the mean systolic blood pressure for women aged between 18
and 24 is 118 mmHg. At the 95% confidence level, the margin of error is 2.4 mmHg. How
would you interpret this margin of error?
7)
8)
A pizza house is considering offering the following special: “Pizza delivered in less than 30
minutes or the pizza is free.” Based on a random sample of 40 delivery times, the manager
obtains the following 99% confidence interval for the true mean delivery time, µ: 25
minutes to 30 minutes. The manufacturer concludes that at least 99% of the pizzas are
delivered in under 30 minutes and that it will be safe to go ahead with the special offer. Is
the manager’s interpretation correct? Why or why not? Explain your thinking.
8)
9)
Describe the steps involved in using the z–interval procedure to obtain a confidence
interval for the population mean when is known. Include a description of the
preliminary data analysis. Why is preliminary data analysis needed?
9)
10)
Suppose that you wish to obtain a 95% confidence interval for a population mean. The
population is normally distributed, the sample size is 20, and the population standard
deviation is unknown. The correct procedure to use is the t–interval procedure. If you
mistakenly use the z–interval procedure, will the resulting confidence interval be too wide
or too narrow? Will the true confidence level associated with this interval be greater than
or less than 95%?
10)
11)
Under what conditions would you choose to use the t–interval procedure instead of the
z–interval procedure in order to obtain a confidence interval for a population mean? What
conditions must be satisfied in order to use the t–interval procedure?
11)
12)
If the sample size is small (less than 15), under what conditions is it reasonable to use the
z–interval procedure to obtain a confidence interval for the population mean? If the
sample size is moderate (between 15 and 30), under what conditions is it reasonable to use
the z–interval procedure to obtain a confidence interval for the population mean?
12)
13)
A psychologist wishes to estimate the mean IQ of students at one college. He picks a
random sample of 49 students from the college, measures each student’s IQ, and obtains
the following 95% confidence interval for the population mean, µ: 115 to 125. How would
you interpret this confidence interval?
13)
14)
A sample mean is used to estimate a population mean. To obtain a margin of error of 1.5 at
a confidence level of 95%, a sample size of 120 is needed. Would the required sample size
be larger or smaller if the researcher wished to
(a) increase the confidence level while keeping the same margin of error?
(b) decrease the margin of error while keeping the same confidence level?
Explain your answers.
14)
15)
Based on a sample of size 25, a researcher obtains an estimate of 64.4 inches for the mean
height of all women aged 30–40. At the 95% confidence level, the margin of error is
1.2 inches. Do you agree with the interpretation below? If not, explain why not and give a
correct interpretation.
Researcher’s interpretation: “We can be 95% confident that the height of a randomly
selected woman will differ from 64.4 inches by at most 1.2 inches.”
15)
16)
A coffee manufacturer claims that the mean amount of coffee in its 4–ounce jars is
4.1 ounces. Based on a sample of 50 jars, a consumer advocate group obtains the following
99% confidence interval for the population mean weight, µ: 3.95 ounces to 4.05 ounces.
Based on this interval, do you think that the manufacturer’s claim is plausible? Possible?
Explain your thinking.
16)
17)
Rank the following t– and z–values in order from smallest to largest.
(a) z0.05 (b) t0.05 (c) t0.05 (d) t0.95 (e) t0.01
The degrees of freedom in (b), (d), (e) are identical. The degrees of freedom in (c) are greater
than those in (b), (d), (e).
17)
18)
Define margin of error. Explain the relation between the confidence interval and the
margin of error. Suppose a confidence interval is 9.65 <µ< 11.35. Find the sample mean x
and the margin of error E.
18)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
19)
A confidence interval for a population mean has a margin of error of 4. Determine the length of the
confidence interval.
19)
A)
16
B)
2
C)
8
D)
4
Find the specified t–value.
20)
For a t–curve with df = 24, find t0.005 .
20)
A)
1.711
B)
2.807
C)
2.797
D)
2.492
Find the necessary sample size.
21)
Weights of women in one age group are normally distributed with a standard deviation of 10 lb.
A researcher wishes to estimate the mean weight of all women in this age group. Find how large a
sample must be drawn in order to be 90 percent confident that the sample mean will not differ from
the population mean by more than 3.4 lb.
21)
A)
24
B)
34
C)
22
D)
25
5
22)
The weekly earnings of students in one age group are normally distributed with a standard
deviation of 47 dollars. A researcher wishes to estimate the mean weekly earnings of students in
this age group. Find the sample size needed to assure with 95 percent confidence that the sample
mean will not differ from the population mean by more than 5 dollars.
22)
A)
8
B)
340
C)
37
D)
6
Find the specified t–value.
23)
For a t–curve with df = 15, find t0.025 .
23)
A)
2.120
B)
1.960
C)
2.131
D)
2.145
Solve the problem.
24)
A confidence interval for a population mean has a margin of error of 0.09. Determine the length of
the confidence interval.
24)
A)
0.18
B)
0.008
C)
0.045
D)
0.09
Find the confidence interval specified.
25)
49 packages are randomly selected from packages received by a parcel service. The sample has a
mean weight of 28.6 pounds. Assume that =3.1 pounds. What is the 95% confidence interval for
the true mean weight, µ, of all packages received by the parcel service?
25)
A)
27.6 to 29.6 pounds
B)
27.5 to 29.7 pounds
C)
27.9 to 29.3 pounds
D)
27.7 to 29.5 pounds
Provide an appropriate response.
26)
Suppose you have obtained a 95% confidence interval for µ. Which of the following statements
is/are true regarding the relationship between precision and confidence level? Assume that the
sample size is fixed.
A. Increasing the confidence level to 99% will result in a narrower interval.
B. Decreasing the confidence level to 90% will result in greater precision.
C. Decreasing the precision will result in a higher confidence level.
D. Increasing the precision will result in a higher confidence level.
26)
A)
A and D
B)
B and C
C)
A and C
D)
B and D
Find the requested value.
27)
A researcher for a car insurance company wishes to estimate the mean annual premium that men
aged 20–24 pay for their car insurance. A random sample of 16 men aged between 20 and 24 yields
the following annual premiums, in dollars.
676 762 691 718
522 654 405 504
780 474 580 725
856 610 720 985
Use the data to obtain a point estimate of the mean annual premium for all men aged between 20
and 24. Round your answer to the nearest dollar.
27)
A)
$650
B)
$646
C)
$658
D)
$666
Find the specified t–value.
28)
For a t–curve with df = 11, find t0.10 .
28)
A)
1.280
B)
2.718
C)
1.372
D)
1.363
Find the confidence interval specified.
29)
A laboratory tested 87 chicken eggs and found that the mean amount of cholesterol was 233
milligrams. Assume that =18.6 milligrams. Construct a 95% confidence interval for the true mean
cholesterol content, µ, of all such eggs.
29)
A)
229.7 to 236.3 milligrams
B)
227.9 to 238.1 milligrams
C)
228.4 to 237.6 milligrams
D)
229.1 to 236.9 milligrams
Find the specified t–value.
30)
For a t–curve with df = 6, find the two t–values that divide the area under the curve into a middle
0.99 area and two outside areas of 0.005.
30)
A)
0, 3.707
B)
–3.143, 3.143
C)
0, 3.143
D)
–3.707, 3.707
31)
For a t–curve with df = 27, find the two t–values that divide the area under the curve into a middle
0.95 area and two outside areas of 0.025.
31)
A)
0, 2.052
B)
–1.703, 1.703
C)
0, 1.703
D)
–2.052, 2.052
Find the requested value.
32)
A long–distance phone company wishes to estimate the mean duration of long–distance calls
originating in California. A random sample of 15 long–distance calls originating in California
yields the following call durations, in minutes.
3 3 2 4 5
39 39 25 31 33
1 19 12 2 37
Use the data to obtain a point estimate of the mean call duration for all long–distance calls
originating in California.
32)
A)
17.0 minutes
B)
17.6 minutes
C)
17.2 minutes
D)
18.0 minutes
Find the specified t–value.
33)
For a t–curve with df = 4, find t0.05 .
33)
A)
1.645
B)
4.604
C)
2.132
D)
2.353
Provide an appropriate response.
34)
Is it true that the point estimate of a population mean must lie within the range of values defined
by the corresponding confidence–interval estimate, regardless of the level of confidence achieved?
Explain.
34)
A)
Yes. By definition, the prescribed confidence interval contains the value of the point estimate.
B)
No. The confidence interval only defines a range of values that is likely to contain the point
estimate with some prescribed level of confidence. This range of values is not guaranteed to
contain the point estimate.
35)
Suppose you have obtained a confidence interval for µ, but wish to obtain a greater degree of
precision. Which of the following would result in a narrower confidence interval?
A. Increasing the sample size while keeping the confidence level fixed
B. Decreasing the sample size while keeping the confidence level fixed
C. Increasing the confidence level while keeping the sample size fixed
D. Decreasing the confidence level while keeping the sample size fixed
35)
A)
B and D
B)
A and D
C)
B and C
D)
A and C
36)
Suppose that you wish to obtain a confidence interval for a population mean. Under the conditions
described below, should you use the z–interval procedure, the t–interval procedure, or neither?
– The population standard deviation is known.
– The population is far from being normally distributed.
– the sample size is large.
36)
A)
z–interval procedure
B)
Neither
C)
t–interval procedure
Find the requested confidence interval.
37)
A survey revealed the amount of pocket money held by each student in a sample of students from
Statistics 205. The amounts (in dollars) are 5.50, 10.20, 3.15, 6.08, 4.85, 4.80, 4.70, 8.52, 7.90, 6.10, 3.50,
5.40, 7.10, 9.60, 6.20, and 4.28. Assuming = 2.00, find the 99.74% confidence interval for the
population mean.
37)
A)
$0 to $4.00
B)
$4.62 to $7.62
C)
$5.62 to $6.62
D)
$4.12 to $8.12
Find the necessary sample size.
38)
Scores on a certain test are normally distributed with a variance of 22. A researcher wishes to
estimate the mean score achieved by all adults on the test. Find the sample size needed to assure
with 95 percent confidence that the sample mean will not differ from the population mean by more
than 2 units.
38)
A)
10
B)
43
C)
465
D)
22
Find the confidence interval specified. Assume that the population is normally distributed.
39)
A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 207
milligrams with s =18.1 milligrams. Construct a 95% confidence interval for the true mean
cholesterol content of all such eggs.
39)
A)
195.6 to 218.4 milligrams
B)
195.4 to 218.6 milligrams
C)
195.5 to 218.5 milligrams
D)
197.6 to 216.4 milligrams
Provide an appropriate response.
40)
Given that the confidence level is x%, give an expression in terms of x for .
40)
A)
x
100
B)
100 – x
C)
100 – (1 – x)
D)
1 –x
100
41)
Suppose that you wish to obtain a confidence interval for a population mean. Under the conditions
described below, should you use the z–interval procedure, the t–interval procedure, or neither?
– The population standard deviation is unknown.
– The population is normally distributed.
– The sample size is small.
41)
A)
z–interval procedure
B)
t–interval procedure
C)
Neither
Find the specified t–value.
42)
For a t–curve with df = 20, find t0.01 .
42)
A)
1.325
B)
2.330
C)
2.539
D)
2.528
Find the confidence interval specified. Assume that the population is normally distributed.
43)
The football coach randomly selected ten players and timed how long each player took to perform
a certain drill. The times (in minutes) were:
8.3 7.3 13.8 12.6 11.1
9.5 13.7 14.2 9.8 12.6
Determine a 95% confidence interval for the mean time for all players.
43)
A)
13.00 to 9.60 minutes
B)
9.70 to 12.90 minutes
C)
9.60 to 13.00 minutes
D)
12.90 to 9.70 minutes
Solve the problem.
44)
A sample of 49 eggs yields a mean weight of 1.69 ounces. Assuming that =0.46 ounces, find the
margin of error in estimating µ at the 95% level of confidence.
44)
A)
0.02 oz
B)
0.47 oz
C)
0.11 oz
D)
0.13 oz
Find the requested value.
45)
Physiologists often use the forced vital capacity as a way to assess a person’s ability to move air in
and out of their lungs. A researcher wishes to estimate the forced vital capacity of people suffering
from asthma. A random sample of 15 asthmatics yields the following data on forced vital capacity,
in liters.
5.3 3.7 3.3 3.0 2.8
2.8 5.2 4.7 5.0 5.2
3.2 3.5 4.8 4.0 5.1
Use the data to obtain a point estimate of the mean forced vital capacity for all asthmatics.
45)
A)
4.03 liters
B)
4.25 liters
C)
4.11 liters
D)
4.19 liters
Provide an appropriate response.
46)
In which of the following situations is it reasonable to use the z–interval procedure to obtain a
confidence interval for the population mean? Assume that the population standard deviation is
known.
A. n = 10, the data contain no outliers, the variable under consideration is not normally
distributed.
B. n = 10, the variable under consideration is normally distributed.
C. n = 18, the data contain no outliers, the variable under consideration is far from being normally
distributed.
D. n = 18, the data contain outliers, the variable under consideration is normally distributed.
46)
A)
A, B, C
B)
C
C)
B
D)
B and D
47)
Find the value of that corresponds to a confidence level of 91.2%.
47)
A)
0.912
B)
8.8
C)
0.088
D)
0.88
Find the requested confidence interval.
48)
The data below consists of the test scores of 32 students. Assuming = 13.36, determine a 95.44%
confidence interval for the population mean.
80 74 61 93 96 70 80 64
51 98 93 87 72 77 84 96
100 67 71 79 99 85 66 70
57 75 86 92 94 65 84 91
48)
A)
69.39 to 83.58
B)
79.49 to 89.39
C)
78.39 to 87.39
D)
75.19 to 84.63
Find the confidence interval specified. Assume that the population is normally distributed.
49)
The principal randomly selected six students to take an aptitude test. Their scores were:
79.5 88.5 83.2 80.2 71.9 75.6
Determine a 90% confidence interval for the mean score for all students.
49)
A)
74.95 to 84.68
B)
84.68 to 74.95
C)
75.05 to 84.58
D)
84.58 to 75.05
Find the confidence interval specified.
50)
A sample of 36 people were randomly selected from among the workers in a shoe factory. The time
taken for each person to polish a finished shoe was measured. The sample mean was 4.2 minutes.
Assume that =0.99 minutes. Construct a 90% confidence interval for the true mean time, µ, to
polish a shoe.
50)
A)
3.93 to 4.47 minutes
B)
3.82 to 4.58 minutes
C)
3.77 to 4.63 minutes
D)
3.88 to 4.52 minutes
51)
30 people are selected randomly from a certain town. If their mean age is 32.2 and =8.5, find a
95% confidence interval for the true mean age, µ, of everyone in the town.
51)
A)
29.17 to 35.23
B)
28.10 to 36.30
C)
30.21 to 34.19
D)
29.16 to 35.24