Deck 13: Confidence Intervals and Hypothesis Tests for Means

Question 1

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 1

t = t = 3.531

Question 2

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?

Answer 2

A Type I error would be concluding that the average number of hours volunteered by the firm's professional employees is more than 15 hours, when in fact it is not. The firm would miss an opportunity to increase community involvement among its professional employees.

Question 3

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

Answer 3

C

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.

What is the associated P-value?

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.

What is the value of the test statistic?

Question 6

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 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.
State your conclusion using α = 0.05.

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.
What is the associated P-value?

Question 9

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 10

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 11

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 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.

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

Question 13

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 14

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 15

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 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.

A Type II error in this context would be

A) failing to detect an increase in the average life expectancy, when in fact it has
B) failing to detect an increase in the average life expectancy, when in fact it hasn't
C) detecting an increase in the average life expectancy, when in fact it has
D) detecting an increase in the average life expectancy, when in fact it hasn't
E) both A and D

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.

Write the null and alternative hypotheses.

Question 18

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 19

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?

Question 20

State your conclusion using α = 0.05.

Question 21

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 22

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 the 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 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 the machine and a technician carefully measures the volume in each cup. In one such test, the following data were observed:

At 5%

A) We fail to reject the null hypothesis and do not adjust the machines because there is not enough evidence to show they are not functioning properly.
B) We reject the null hypothesis.
C) We know the machines are working properly.
D) We know the machines need adjustment.
E) We reject the null hypothesis and adjust the machines because there is enough evidence to show they are not functioning properly.

Question 24

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 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 25

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 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.
The correct value of the test statistic is

A) t = 3.629
B) t = -3.629
C) t = 1.223
D) t = -1.223
E) t = 0.789

Question 27

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 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 we cannot conclude that students work in excess of 20 hours per week because there is not enough evidence to show otherwise.
B) We reject the null hypothesis.
C) We know that students do not work more than 20 hours.
D) We know students work more than 20 hours per week.
E) No conclusion can be reached about the hypothesis with the information that is given.

Question 28

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 need to increase their premiums because there is evidence to indicate an increase in average life expectancy.
C) We fail to reject the null hypothesis.
D) We fail to reject the null hypothesis, and ABI Insurance should not need to increase their premiums because there is little evidence to indicate an increase in average life expectancy.
E) We cannot reach any conclusion about the hypothesis with the given information.

Question 29

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 the 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 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 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 31

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 32

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 the evidence indicates that professional employees volunteer an average of no more than 15 hours per month.
D) We reject the null hypothesis, and the firm shouldn't need to institute an incentive program because the evidence indicates that professional employees volunteer an average of more than 15 hours per month.
E) No conclusion can be reached about the hypothesis with the information that is given.

Question 33

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) 33
D) 25
E) 90

Question 34

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 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 35

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 the 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 36

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 the machine and a technician carefully measures the volume in each cup. In one such test, the following data were observed:

In order to be accurate to 99% 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