Deck 11: Confidence Intervals for Proportions

Question 1

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
The Quality Control Department wants to estimate the true percentage of rework for electrical components to within 4%, with 99% confidence. How many components should they sample?

Answer 1

n = (2.576)2 (.12) (.88) / (.04)2 = 437.96 = 438 electrical components

Question 2

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
They actually select a random sample of 450 electrical components and find that 46 of those had to be reworked. Create the 99% confidence interval.

Answer 2

Confidence interval: ±2.576 SE ( ) = 0.10222±0.03679, i.e. between 0.0654 and 0.1390

Question 3

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
They actually select a random sample of 525 employment records, and find that 229 of the people are females. Construct the 90% confidence interval.

Answer 3

Confidence interval: ± 1.645 SE ( ) = 0.43619 ± 0.03560, i.e. between 0.4006 and 0.4718

Question 4

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
The 95% confidence interval based on this data is 0.0742 to 0.1302. Which of the following is the correct interpretation?

A) The percentage of electronic components that are to be reworked is between 7.42% and 13.02%.
B) We are 95% confident that between 7.42% and 13.02% of electrical components are to be reworked.
C) The margin of error for the true percentage of electrical components that are reworked is between 7.42% and 13.02%.
D) All samples of size 450 will yield a percentage of reworked electrical components that falls within 7.42% and 13.02%.
E) The probability that the true number of electrical parts that have to be reworked is between 7.42% and 13.02% is 0.05.

Question 5

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
The Quality Control Department wants to estimate the true percentage of rework for electrical components to within 4%, with 99% confidence. How many components should they sample?

A) 651
B) 1000
C) 344
D) 438
E) 579

Question 6

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
Based on the 95% confidence interval, should the Quality Control Department conclude that the percentage of rework for the electrical components is lower than the rate of 12% for non-electrical components?

A) Yes, because the lower limit of the confidence interval is 7.42%.
B) Yes, because 12% is contained within the 95% confidence interval.
C) No, because 12% is contained within the 95% confidence interval.
D) Yes, because the upper limit of the confidence interval is 13.02%.
E) Yes, because 12 is larger than the width of the confidence interval.

Question 7

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
Statistics Canada wants to estimate the percentage of females in the Canadian labour force to within 5% with 90% confidence. How many employment records should be sampled?

A) 121
B) 269
C) 451
D) 382
E) 1000

Question 8

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
Interpret the confidence interval in this context.

Question 9

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
They actually select a random sample of 525 employment records, and find that 229 of the people are females. The 90% confidence interval is

A) 0.4006 to 0.4718.
B) 0.2747 to 0.5973.
C) 0.1776 to 0.6944.
D) 0.4235 to 0.5679.
E) 0.1243 to 0.7100.

Question 10

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
Interpret the confidence interval in this context.

Question 11

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
Statistics Canada wants to estimate the percentage of females in the Canadian labour force to within 5%, with 90% confidence. How many employment records should be sampled?

Question 12

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
Explain what 90% confidence means in this context.

Question 13

Are the assumptions and conditions for constructing a confidence interval met? Explain.

Question 14

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
They actually select a random sample of 450 electrical components and find that 46 of those had to be reworked. The 99% confidence interval is

A) 0.0654 to 0.1390.
B) 0.0432 to 0.1608.
C) 0.0763 to 0.1277.
D) 0.0541 to 0.1499.
E) 0.0787 to 0.1257.

Question 15

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
Explain what 99% confidence means in this context.

Question 16

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
Should representatives from Statistics Canada conclude that the percentage of females in the Canadian labour force is lower than European rate of 46%? Explain.

Question 17

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
Suppose that Statistics Canada wants to be 90% confident of estimating the percentage of females in the labour force to within 2% of the true percentage. To do this they would have to

A) decrease the sample size.
B) select the same number of employment records.
C) increase the sample size.
D) increase the margin of error.
E) increase the sampling error.

Question 18

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
Should the Quality Control Department conclude that the percentage of rework for the electrical components is lower than the rate of 12% for non-electrical components? Explain.

Question 19

Consider the following to answer the question(s) below:
One division of a large defense contractor manufactures telecommunication equipment for the military. This division reports that 12% of non-electrical components are reworked. Management wants to determine if this percentage is the same as the percentage rework for electrical components manufactured by the entire company. The Quality Control Department plans to check a random sample of the over 10,000 electrical components manufactured across all divisions.
Are the assumptions and conditions for constructing a confidence interval met? Explain.

Question 20

Consider the following to answer the question(s) below:
EU (European Union) countries report that 46% of their labour force is female. The United Nations wants to determine if the percentage of females in the Canadian labour force is the same. Representatives from Statistics Canada plan to check a random sample of over 10,000 employment records on file to estimate the percentage of females in the Canadian labour force.
If Statistics Canada wishes to tighten its interval, they should

A) increase the confidence level.
B) increase the sample size.
C) decrease the sample size.
D) increase the margin of error.
E) double the margin of error .

Question 21

City officials want to estimate the proportion of visitors who are repeat visitors to the Rideau Canal Skateway. They take a sample of 400 visitors and find that 100 are on a second visit. The 95% confidence interval is

A) 0.2076 to 0.292.
B) 0.1942 to 0.3058.
C) 0.2144 to 0.2856.
D) 0.1996 to 0.3004.
E) 0.2222 to 0.2777.

Question 22

All else being equal, increasing the level of confidence desired will

A) tighten the confidence interval.
B) decrease the margin of error.
C) leave the margin of error unchanged.
D) increase the margin of error.
E) increase the standard error.

Question 23

A study was recently conducted at a major university to estimate the difference in the proportion of business school and non-business school graduates who go on to a higher degree program within five years of graduation A random sample of 400 business school graduates showed that 75 had gone to graduate school, while in a random sample of 500 non-business graduates, 137 had done so. The standard error for the difference in the proportions of the two groups is

A) 0.0279.
B) 0.
C) 0.0008.
D) 0.0865.
E) 0.1875.

Question 24

A sample of 250 people resulted in a confidence interval estimate of between 0.14 and 0.20 for the proportion of people who believe that tax increase proposed by the federal government is justified. Based on this information, what was the confidence level used in this estimation?

A) approximately 79%
B) 95%
C) approximately 36.9%
D) approximately 1.26%
E) approximately 90%

Question 25

City officials want to estimate the proportion of visitors who are repeat visitors to the Rideau Canal Skateway. They take a sample of 400 visitors and find that 100 are on a second visit. The 99% confidence interval is 0.1942 to 0.3058. The correct interpretation of the interval is

A) we are 99% confident that between 19.42% and 30.58% of visitors are on a second visit.
B) the proportion of repeat visitors is between 19.42% and 30.58%.
C) the probability is 5% that between 19.42% and 30.58% of visitors are on a second visit.
D) the margin of error is between 19.42% and 30.58%.
E) all samples of 400 will have between 19.42% and 30.58% of visitors on a second visit.

Question 26

A study was recently conducted at a major university to estimate the difference in the proportion of business school and non-business school graduates who go on to a higher degree program within five years of graduation. A random sample of 400 business school graduates showed that 75 had gone to graduate school, while in a random sample of 500 non-business graduates, 137 had done so. The 95% confidence interval is

A) -0.1412 to -0.0318.
B) -2.0465 to 1.8375.
C) -0.0880 to -0.0850.
D) -0.1144 to -0.0586.
E) -0.0873 to -0.0857.

Question 27

City officials want to estimate the proportion of visitors who are repeat visitors to the Rideau Canal Skateway. From previous experience they believe the proportion is not larger than 20 percent. They want to estimate the proportion to within 0.04 percentage points with 95 % confidence. The sample size they should use is

A) 385.
B) 601.
C) 97.
D) 10.
E) 248.

Question 28

A human resources manager wishes to estimate the proportion of employees in her large company who have supplemental health insurance. What sample size should she select if she wants 95% confidence and a margin of error of 0.01?

A) 9,604
B) 3,458
C) 6,765
D) 381
E) 4,096

Question 29

Which of the following is not an assumption / condition required for constructing a confidence interval for the proportion?

A) Randomization condition
B) Linearity condition
C) Success/Failure condition
D) 10% condition
E) Independence assumption

Question 30

City officials want to estimate the proportion of visitors who are repeat visitors to the Rideau Canal Skateway. Suppose they find they need a sample size of 15,000 people to achieve a margin of error of 0.01 percentage points with 99 % confidence. This sample is too large to be practical. How can they reduce the sample size?

A) Use a lower confidence level.
B) Use a smaller margin of error.
C) Use a higher level of confidence.
D) Conduct a census.
E) Use a higher confidence level and a smaller margin of error.

Question 31

A financial analyst is interested in estimating the proportion of publicly traded companies on the Toronto Stock Exchange that have cash balances of more than 10 percent of the total assets of the company. A random sample of 100 companies shows that 13 had cash balances of more than 10 percent of assets. The 90 % confidence interval estimate is

A) 0.0747 to 0.1853.
B) 0.0641 to 0.1959.
C) -0.0870 to 0.1730.
D) -0.0507 to 0.1494.
E) -0.0412 to 0.1588.

Question 32

The produce manager at a local Sobeys store is interested in estimating the percentage of apples that arrive on a shipment with bruises. A random sample of 150 apples showed 14 with bruises. Based on this information, what is the margin of error for a 95% confidence interval estimate?

A) 0.0466
B) 0.0933
C) 0.0006
D) 0.0238
E) 0.0392