Deck 13: Confidence Intervals and Hypothesis Tests for Means

Question 1

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

What is the associated P-value?

Answer 1

P-value = 0.063

Question 2

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

For more accurate cost determination, ABI Insurance wants to estimate the average life expectancy to within one year with 95% confidence. How many randomly selected recently paid policies would they need to sample?

Answer 2

n = 77.1 or 78

Question 3

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
What is the value of the test statistic?

Answer 3

t = t = 3.531

Question 4

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

Write the null and alternative hypotheses.

Question 5

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

The correct value of the test statistic is

A) t = -1.597.
B) t = 2.779.
C) t = 1.597.
D) t = 0.443.
E) t = -2.779.

Question 6

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

State your conclusion using α = 0.05.

Question 7

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
In this context, describe the Type I error possible. How might such an error impact the software development firm?

Question 8

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
For a more accurate determination, top management wants to estimate the average number of hours volunteered per month by their professional staff to within one hour with 99% confidence. How many randomly selected professional employees would they need to sample?

Question 9

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
State your conclusion using α = 0.05.

Question 10

Suppose 40 professional employees are randomly selected. This sample yields a mean of 15.2 hours and a standard deviation of 1.8 hours. Find the 95% confidence interval and interpret.

Question 11

Suppose ABI samples 100 recently paid policies. This sample yields a mean of 77.7 years and a standard deviation of 3.6 years. Find a 90% confidence interval and interpret.

Question 12

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

The correct null and alternative hypotheses are

A) H0 : µ = 77 and HA : µ > 77.
B) H0 : µ = 77 and HA : µ < 77.
C) H0 : µ = 77 and HA : µ ≠ 77.
D) H0 : µ ≠ 77 and HA : µ = 77.
E) H0 : µ > 77 and HA : µ = 77

Question 13

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
In this context, describe the Type II error possible. How might such an error impact the software development firm?

Question 14

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

The P-value associated with the test statistic is

A) 0.025.
B) 0.228.
C) 0.01.
D) 0.05.
E) 0.063.

Question 15

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

In this context, describe the Type II error possible. How might such an error impact ABI Insurance?

Question 16

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

In this context, describe the Type I error possible. How might such an error impact ABI Insurance?

Question 17

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

What is the value of the test statistic?

Question 18

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
Write the null and alternative hypotheses.

Question 19

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
What is the associated P-value?

Question 20

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

A Type II error in this context would be

A) failing to detect that the average life expectancy has increased, when in fact it has.
B) failing to detect that the average life expectancy has increased, when in fact it is unchanged.
C) detecting that the average life expectancy has increased, when in fact it hasn't.
D) failing to detect that the average life expectancy is unchanged.
E) detecting that the average life expectancy is unchanged.

Question 21

Suppose 40 professional employees are randomly selected. This sample yields a mean of 15.2 hours and a standard deviation of 1.8 hours. The 95% confidence interval is

A) 12.45 to 17.85 hours.
B) 14.62 to 15.78 hours.
C) 13.88 to 14.29 hours.
D) 10.65 to 20.22 hours.
E) 11.98 to 18.02 hours.

Question 22

Consider the following to answer the question(s) below:
The cost of a university education has increased at a much faster rate than costs in general over the past twenty years. In order to compensate for this, many students work part- or full-time in addition to attending classes. At one university, it is believed that the average number of hours students work per week exceeds 20. A random sample of 20 students was selected and the following values were observed:

The correct value of the test statistic is

A) 0.1314.
B) 3.04.
C) 2.0930.
D) 1.7291.
E) 0.8968.

Question 23

Consider the following to answer the question(s) below:
The Dispenser Right Company makes a soft drink dispensing machine used in many fast food restaurants. When the machine is running properly, the average number of fluid ounces in the cup should be 14. Periodically the machines need to be tested to make sure that they have not gone out of adjustment. To do this, six cups are filled by a particular machine and a technician carefully measures the volume in each cup. In one such test, the following data were observed:

The correct null and alternative hypotheses are

A) H0 : µ = 14 and HA : µ ≠ 14.
B) H0 : µ ≠ 14 and HA : µ = 14.
C) H0 : µ = 14 and HA : µ > 14.
D) H0 : µ = 14 and HA : µ < 14.
E) H0 : µ < 14 and HA : µ > 14.

Question 24

Consider the following to answer the question(s) below:
The Dispenser Right Company makes a soft drink dispensing machine used in many fast food restaurants. When the machine is running properly, the average number of fluid ounces in the cup should be 14. Periodically the machines need to be tested to make sure that they have not gone out of adjustment. To do this, six cups are filled by a particular machine and a technician carefully measures the volume in each cup. In one such test, the following data were observed:

The correct value of the test statistic is

A) 0.2889.
B) -0.2889.
C) 0.1837.
D) 0.0750.
E) 0.7842.

Question 25

Consider the following to answer the question(s) below:
The Dispenser Right Company makes a soft drink dispensing machine used in many fast food restaurants. When the machine is running properly, the average number of fluid ounces in the cup should be 14. Periodically the machines need to be tested to make sure that they have not gone out of adjustment. To do this, six cups are filled by a particular machine and a technician carefully measures the volume in each cup. In one such test, the following data were observed:

The 99% confidence interval is

A) 13.72 to 14.32 ounces.
B) 13.83 to 14.21 ounces.
C) 13.87 to 14.71 ounces.
D) 13.70 to 14.30 ounces.
E) 13.81 to 14.19 ounces.

Question 26

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
For a more accurate determination, top management wants to estimate the average number of hours volunteered per month by their professional staff to within one hour with 99% confidence. How many randomly selected professional employees would they need to sample?

A) 56
B) 120
C) 37
D) 25
E) 90

Question 27

Consider the following to answer the question(s) below:
The Dispenser Right Company makes a soft drink dispensing machine used in many fast food restaurants. When the machine is running properly, the average number of fluid ounces in the cup should be 14. Periodically the machines need to be tested to make sure that they have not gone out of adjustment. To do this, six cups are filled by a particular machine and a technician carefully measures the volume in each cup. In one such test, the following data were observed:

In order to be 99% accurate to within 0.02 ounces, what sample size would be needed if the sample standard deviation is 0.18?

A) 538
B) 23
C) 11
D) 1159
E) 440

Question 28

Consider the following to answer the question(s) below:
The cost of a university education has increased at a much faster rate than costs in general over the past twenty years. In order to compensate for this, many students work part- or full-time in addition to attending classes. At one university, it is believed that the average number of hours students work per week exceeds 20. A random sample of 20 students was selected and the following values were observed:

The 95% confidence interval is

A) 14.03 to 26.77 hours.
B) 11.69 to 29.11 hours.
C) 15.14 to 25.66 hours.
D) 12.67 to 28.13 hours.
E) 13.63 to 26.37 hours.

Question 29

Consider the following to answer the question(s) below:
The cost of a university education has increased at a much faster rate than costs in general over the past twenty years. In order to compensate for this, many students work part- or full-time in addition to attending classes. At one university, it is believed that the average number of hours students work per week exceeds 20. A random sample of 20 students was selected and the following values were observed:

For a more accurate determination, the university wants to estimate the average number of hours worked per week by students to within one hour with 99% confidence. How many randomly selected students would they need to sample?

A) 1229
B) 35
C) 554
D) 24
E) 62

Question 30

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
The correct value of the test statistic is

A) t = 3.531.
B) t = -3.531.
C) t = 1.223.
D) t = -1.223.
E) t = 0.789.

Question 31

Consider the following to answer the question(s) below:
The cost of a university education has increased at a much faster rate than costs in general over the past twenty years. In order to compensate for this, many students work part- or full-time in addition to attending classes. At one university, it is believed that the average number of hours students work per week exceeds 20. A random sample of 20 students was selected and the following values were observed:

The correct null and alternative hypotheses are

A) H0 : µ = 20 and HA : µ > 20.
B) H0 : µ = 20 and HA : µ ≠ 20.
C) H0 : µ = 20 and HA : µ < 20.
D) H0 : µ > 20 and HA : µ = 20.
E) H0 : µ < 20 and HA : µ > 20.

Question 32

Consider the following to answer the question(s) below:
The cost of a university education has increased at a much faster rate than costs in general over the past twenty years. In order to compensate for this, many students work part- or full-time in addition to attending classes. At one university, it is believed that the average number of hours students work per week exceeds 20. A random sample of 20 students was selected and the following values were observed:

At α = 0.05

A) we fail to reject the null hypothesis, and there is no evidence to suggest that students work more than 20 hours per week.
B) we reject the null hypothesis.
C) we support the alternative hypothesis; there is strong evidence to suggest that students work more than 20 hours per week.
D) we know students work more than 20 hours per week, on average.
E) we reject the null hypothesis. The P-value = 0.4884 indicates that students work more than 20 hours per week.

Question 33

At 5%,

A) we fail to reject the null hypothesis. They should not adjust the machine because there is no evidence to suggest that it is not functioning properly.
B) we reject the null hypothesis. The P-value is very low.
C) we support the alternative hypothesis because we know the machine is working properly.
D) we support the alternative hypothesis because we know the machine needs adjustment.
E) we reject the null hypothesis. The company should adjust the machine because there is strong evidence to show it is not functioning properly.

Question 34

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
At α = 0.05,

A) we reject the alternative hypothesis.
B) we fail to reject the null hypothesis.
C) the firm should institute an incentive program because there is evidence indicating that professional employees volunteer less than 15 hours per month on average.
D) we reject the null hypothesis, and the firm shouldn't institute an incentive program because there is strong evidence indicating that professional employees volunteer more than 15 hours per month, on average.
E) we reject HA : µ > 15. The P-value indicates that professional employees volunteer exactly 15 hours per month.

Question 35

Consider the following to answer the question(s) below:
A large software development firm recently relocated its facilities. Top management is interested in fostering good relations with their new local community and has encouraged their professional employees to engage in local service activities. They believe that the firm's professionals volunteer an average of more than 15 hours per month. If this is not the case, they will institute an incentive program to increase community involvement. A random sample of 24 professionals reported the following number of hours:
The sample has a mean of 16.6 hours and a standard deviation of 2.22 hours.
The correct null and alternative hypotheses are

A) H0 : µ = 15 and HA : µ < 15.
B) H0 : µ = 15 and HA : µ > 15.
C) H0 : µ = 15 and HA : µ ≠ 15.
D) H0 : µ ≠ 15 and HA : µ = 15.
E) H0 : µ < 15 and HA : µ > 15.

Question 36

Consider the following to answer the question(s) below:
Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Last year the average life expectancy of all policyholders was 77 years. ABI Insurance wants to determine if their clients now have a longer life expectancy, on average, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample has a mean of 78.6 years and a standard deviation of 4.48 years.

At α = 0.05,

A) we reject the null hypothesis.
B) ABI Insurance should increase their premiums because there is evidence indicating an increase in average life expectancy.
C) we fail to reject the alternative hypothesis HA : µ > 77.
D) we fail to reject the null hypothesis, and ABI Insurance should not increase their premiums because there is no evidence indicating an increase in average life expectancy.
E) we support the alternative hypothesis HA : µ > 77. ABI Insurance should increase their premiums.