Deck 8: Confidence Intervals for One Population Mean

Question 1

Write the word or phrase that best completes each statement or answers the question.
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?

Answer 1

Preliminary data analysis is needed to verify
that the conditions for using the z-interval procedure are satisfied. This involves
constructing a graph such as a normal probability plot, a boxplot, a stem-and-leaf
plot, or a histogram and examining the data. If it is reasonable to proceed, calculate
the sample mean and obtain the confidence interval by using the formula

Question 2

Write the word or phrase that best completes each statement or answers the question.
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?

Answer 2

The variable in (b)has more variation since its variation is due to the variation of
both the sample means and the sample standard deviations. The variation of the
variable in (a)is due solely to the variation of the sample means. The variable in (a)
has a standard normal distribution while the variable in (b)has a t-distribution with
34 degrees of freedom.

Question 3

Provide an appropriate response.
When estimating a population mean by a sample mean, which of the following does the margin of error not depend on?

A)The sample size
B)The confidence level
C)The population standard deviation
D)The sample mean

Answer 3

D

Question 4

Write the word or phrase that best completes each statement or answers the question.
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, 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.

Question 5

Write the word or phrase that best completes each statement or answers the question.
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?

Question 6

Write the word or phrase that best completes each statement or answers the question.
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?

Question 7

Write the word or phrase that best completes each statement or answers the question.
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.

Question 8

Write the word or phrase that best completes each statement or answers the question.
Rank the following t- and z-values in order from smallest to largest. The degrees of freedom in (b), (d), (e)are identical. The degrees of freedom in (c)are
greater than those in (b), (d), (e).

Question 9

Write the word or phrase that best completes each statement or answers the question.

Question 10

Find the requested value.
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. Use the data to obtain a point estimate of the mean call duration for all long-distance calls
Originating in California.

A)14.4 minutes
B)15.0 minutes
C)14.2 minutes
D)14.7 minutes

Question 11

Find the specified t-value.
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.

A)0, 2.052
B)-2.052, 2.052
C)-1.703, 1.703
D)0, 1.703

Question 12

Write the word or phrase that best completes each statement or answers the question.
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, Explain your thinking.

Question 13

Write the word or phrase that best completes each statement or answers the question.
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, : 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.

Question 14

Write the word or phrase that best completes each statement or answers the question.
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."

Question 15

Write the word or phrase that best completes each statement or answers the question.
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%?

Question 16

Write the word or phrase that best completes each statement or answers the question.
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?

Question 17

Write the word or phrase that best completes each statement or answers the question.
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?

Question 18

Write the word or phrase that best completes each statement or answers the question.
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?

Question 19

Write the word or phrase that best completes each statement or answers the question.

Question 20

Provide an appropriate response.
In stating a confidence-interval estimate of a population mean, the level of confidence increases as the size of the interval .

A)decreases
B)increases

Question 21

Find the confidence interval specified.
49 packages are randomly selected from packages received by a parcel service. The sample has a mean weight of 11.7 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?

A)10.6 to 12.8 pounds
B)11.0 to 12.4 pounds
C)10.7 to 12.7 pounds
D)10.8 to 12.6 pounds

Question 22

Solve the problem.
A sample of 55 eggs yields a mean weight of 1.76 ounces. Assuming that = 0.28 ounces, find the margin of error in estimating µ at the 95% level of confidence.

A)0.06 oz
B)0.01 oz
C)0.47 oz
D)0.07 oz

Question 23

Find the specified t-value.

Question 24

Find the confidence interval specified.
A sample of 38 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 4.0 minutes.
Assume that 0.71 minutes. Construct a 90% confidence interval for the true mean time, to
Polish a shoe.

A)3.70 to 4.30 minutes
B)3.81 to 4.19 minutes
C)3.73 to 4.27 minutes
D)3.77 to 4.23 minutes

Question 25

Provide an appropriate response.
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 not normally distributed.
- The sample size is 12.

A)t-interval procedure
B)Neither
C)z-interval procedure

Question 26

Solve the problem.
Based on a sample of 36 randomly selected years, a 90% confidence interval for the mean annual precipitation in one city is from 46.8 inches to 49.2 inches. Find the margin of error.

A)0.33
B)2.4 inches
C)1.2 inches
D)There is not enough information to find the margin of error.

Question 27

Find the necessary sample size.
The monthly earnings of a group of business students are are normally distributed with a standard deviation of 545 dollars. A researcher wants to estimate the mean monthly earnings of all business
Students. Find the sample size needed to have a confidence level of 95% and a margin of error of
128 dollars.

A)2
B)5
C)61
D)70

Question 28

Find the specified t-value.

Question 29

Find the specified t-value.
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.

A)0, 3.143
B)-3.707, 3.707
C)-3.143, 3.143
D)0, 3.707

Question 30

Solve the problem.
A confidence interval for a population mean has a margin of error of 6.4. Determine the length of the confidence interval.

A)6.4
B)40.96
C)12.8
D)3.2

Question 31

Find the confidence interval specified. Assume that the population is normally distributed.
The football coach randomly selected ten players and timed how long each player took to perform a certain drill. The times (in minutes)were: Determine a 95% confidence interval for the mean time for all players.

A)12.80 to 9.40 minutes
B)9.50 to 12.70 minutes
C)12.70 to 9.50 minutes
D)9.40 to 12.80 minutes

Question 32

Find the confidence interval specified. Assume that the population is normally distributed.
A laboratory tested twelve chicken eggs and found that the mean amount of cholesterol was 188 milligrams with s = 12.7 milligrams. Construct a 95% confidence interval for the true mean
Cholesterol content of all such eggs.

A)179.8 to 196.2 milligrams
B)181.4 to 194.6 milligrams
C)179.9 to 196.1 milligrams
D)180.0 to 196.0 milligrams

Question 33

Solve the problem.
A confidence interval for a population mean, µ, has l ength 179.1. Find the margin of error.

A)89.55
B)358.2
C)32,076.81
D)179.1

Question 34

Provide an appropriate response.
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.

Question 35

Find the specified t-value.
For a t-curve with df = 18, find the t-value having area 0.05 to its right.

A)2.878
B)1.740
C)2.101
D)1.734

Question 36

Find the confidence interval specified.

Question 37

Solve the problem.
A sample of 35 washing machines yields a mean replacement time of 9.0 years. Assuming that = 2.4 years, find the margin of error in estimating at the 90% level of confidence.

A)0.7 years
B)0.5 years
C)2.5 years
D)0.1 years

Question 38

Find the specified t-value.

Question 39

Find the confidence interval specified. Assume that the population is normally distributed.
The principal randomly selected six students to take an aptitude test. Their scores were: Determine a 90% confidence interval for the mean score for all students.

A)74.85 to 87.89
B)87.89 to 74.85
C)74.95 to 87.79
D)87.79 to 74.95

Question 40

Provide an appropriate response.

Question 41

Find the confidence interval specified. Assume that the population is normally distributed.
Thirty randomly selected students took the calculus final. If the sample mean was 93 and the standard deviation was 13.3, construct a 99% confidence interval for the mean score of all students.

A)88.87 to 97.13
B)86.31 to 99.69
C)87.02 to 98.98
D)86.33 to 99.67

Question 42

Provide an appropriate response.
Suppose you have obtained a 95% confidence interval for 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.

A)A and C
B)A and D
C)B and C
D)B and D

Question 43

Provide an appropriate response.
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.

A)Neither
B)t-interval procedure
C)z-interval procedure

Question 44

Find the specified t-value.

Question 45

Find the specified t-value.

Question 46

Find the confidence interval specified. Assume that the population is normally distributed.
A random sample of 30 households was selected from a particular neighborhood. The number of cars for each household is shown below. Estimate the mean number of cars per household for the
Population of households in this neighborhood. Give the 95% confidence interval.

A)1.3 to 1.7 cars
B)1.1 to 1.9 cars
C)1 to 2 cars
D)1.5 to 1.9 cars

Question 47

Provide an appropriate response.
Which of the following statements regarding t-curves is/are true? A. The total area under a t-curve with 10 degrees of freedom is greater than the area under the
Standard normal curve.
B. The t-curve with 10 degrees of freedom is flatter and wider than the standard normal curve.
C. The t-curve with 10 degrees of freedom more closely resembles the standard normal curve than
The t-curve with 20 degrees of freedom.

A)B, C
B)C
C)A
D)B

Question 48

Find the requested value.
A researcher wishes to estimate the mean resting heart rate for long-distance runners. A random sample of 12 long-distance runners yields the following heart rates, in beats per minute. Use the data to obtain a point estimate of the mean resting heart rate for all long distance runners.

A)68.8 beats per minute
B)65.6 beats per minute
C)70.6 beats per minute
D)67.2 beats per minute

Question 49

Provide an appropriate response.
Find the confidence level that corresponds to a value of α of 0.12.

A)12%
B)0.06%
C)88%
D)0.88%

Question 50

Find the necessary sample size.
The weekly earnings of students in one age group are normally distributed with a standard deviation of 10 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 2 dollars.

A)10
B)7
C)20
D)97

Question 51

Find the requested confidence interval.

Question 52

Provide an appropriate response.
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.

A)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.
B)Yes. By definition, the prescribed confidence interval contains the value of the point estimate.

Question 53

Provide an appropriate response.
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 data contain outliers.
- The sample size is small.

A)z-interval procedure
B)t-interval procedure
C)Neither

Question 54

Find the confidence interval specified.

Question 55

Find the confidence interval specified. Assume that the population is normally distributed.
A sociologist develops a test to measure attitudes about public transportation, and 27 randomly selected subjects are given the test. Their mean score is 76.2 and their standard deviation is 21.4.
Construct the 95% confidence interval for the mean score of all such subjects.

A)69.2 to 83.2
B)67.7 to 84.7
C)64.2 to 88.2
D)74.6 to 77.8

Question 56

Find the confidence interval specified.

Question 57

Find the requested confidence interval.

Question 58

Find the confidence interval specified.

Question 59

Find the confidence interval specified.
A random sample of 149 full-grown lobsters had a mean weight of 15 ounces. Assume that = 3.6 ounces. Construct a 95% confidence interval for the population mean

A)14.3 to 15.7 ounces
B)14.4 to 15.6 ounces
C)14.5 to 15.5 ounces
D)14.2 to 15.8 ounces

Question 60

Solve the problem.

Question 61

Solve the problem.
A confidence interval for a population mean has a margin of error of 0.081. Determine the length of the confidence interval.

A)0.007
B)0.081
C)0.041
D)0.162

Question 62

Solve the problem.
A confidence interval for a population mean, µ, has length 86. Find the margin of error.

A)43
B)86
C)21.5
D)1849

Question 63

Find the requested value.
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. Use the data to obtain a point estimate of the mean forced vital capacity for all asthmatics.

A)4.18 liters
B)4.32 liters
C)4.27 liters
D)4.10 liters

Question 64

Find the confidence interval specified. Assume that the population is normally distributed.
A savings and loan association needs information concerning the checking account balances of its local customers. A random sample of 14 accounts was checked and yielded a mean balance of
$664.14 and a standard deviation of $297.29. Find a 90% confidence interval for the true mean
Checking account balance for local customers.

A)$523.43 to $804.85
B)$455.65 to $872.63
C)$493.71 to $834.57
D)$492.52 to $835.76

Question 65

Provide an appropriate response.
A confidence interval for a population mean has a margin of error of 2.2. If the sample mean is 51.3, obtain the confidence interval.

A)From 49.1 to 53.5
B)From 51.3 to 53.5
C)From 49.1 to 51.3
D)From 50.2 to 52.4

Question 66

Find the requested confidence interval.
A random sample of 30 long distance runners aged 20-25 was selected from a running club. The resting heart rates (in beats per minute)of the runners are shown below. Estimate the mean resting
Heart rate for the population of long distance runners aged 20-25. Assuming = 8.03, give the
95.44% confidence interval for the population mean.

A)64.10 to 70.10 beats per minute
B)65.13 to 71.01 beats per minute
C)63.66 to 72.48 beats per minute
D)66.60 to 69.54 beats per minute

Question 67

Provide an appropriate response.
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

A)A and D
B)B and D
C)A and C
D)B and C

Question 68

Find the confidence interval specified.
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.

Question 69

Find the requested confidence interval.

Question 70

Provide an appropriate response.
Find the value of that corresponds to a confidence level of 81%.

A)0.19
B)0.81
C)0.019
D)19

Question 71

Find the requested confidence interval.

Question 72

Provide an appropriate response.
Suppose that scores for men on an aptitude test have greater variability than scores for women on the same test. In other words, the population standard deviation is greater for the population of
Men than for the population of women. Based on a sample of size 50, a 95% confidence interval for
The mean score, , of all women has a margin of error of 2.2. Which of the following confidence
Intervals will have a smaller margin of error?
A. A 99% confidence interval for the mean score of women. Sample size = 50
B. A 95% confidence interval for the mean score of women. Sample size = 100
C. A 95% confidence interval for the mean score of men. Sample size = 50

A)C
B)B
C)A
D)A, B, C

Question 73

Find the requested value.
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. 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.

A)$690
B)$694
C)$711
D)$703

Question 74

Find the requested confidence interval.

Question 75

Provide an appropriate response.

Question 76

Find the necessary sample size.
Weights of women in one age group are normally distributed with a standard deviation Ϭ of 14 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 3.0 lb.

A)57
B)60
C)84
D)59

Question 77

Provide an appropriate response.
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.

A)z-interval procedure
B)t-interval procedure
C)Neither

Question 78

Find the specified t-value.

Question 79

Find the specified t-value.

Question 80

Solve the problem.
Based on a sample of size 41, a 95% confidence interval for the mean score of all students, on an aptitude test is from 60 to 66. Find the margin of error.

A)3.0
B)0.92
C)There is not enough information to find the margin of error.
D)6