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More about Confidence Intervals Answers

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A city planner would like to estimate the true mean annual income of all households in the city. She selects a random sample of 50 households and determines that the 99% confidence interval for the true mean annual income of all households in the city to be $42,000–$68,000. Which of these statements is a correct interpretation of the confidence level?

A
Approximately 99% of all households have incomes between $42,000 and $68,000.
B
The city planner can be 99% confident that the true mean annual income for all households in the city is between $42,000 and $68,000.
C
If many random samples of size 50 are selected from all households in the city, approximately 99% of the intervals would capture the true mean household income.
D
Approximately 99% of the sample means, computed from many random samples of size 50 from all households in the city, will fall between approximately $42,000 and $68.000.
2
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The maker of a cell phone screen protector would like to estimate the proportion of customers who file a warranty claim. To do so, they select a random sample of 200 customers and determine that the 96% confidence interval for the true proportion of customers who file a warranty claim to be 0.15 to 0.28. Which of these statements is a correct interpretation of the confidence level?

A
Approximately 96% of customers file a warranty claim.
B
The cell phone screen protector maker can be 96% confident that the interval from 0.15 to 0.28 captures true proportion of customers who file a warranty claim.
C
If many random samples of size 200 are selected from the population of all customers, approximately 96% of the sample proportions of customers who file a warranty claim will be between 0.15 and 0.28.
D
If many random samples of size 200 are selected from the population of all customers, about 96% of the intervals constructed from the samples would capture the true proportion of customers who file a warranty claim.
3

An animal rescue agent wanted to estimate the true proportion of all animals in shelters that are adopted each month. To do so, she selects a random sample of 100 animals and determines that the 95% confidence interval for the true proportion of animals adopted each month is between 0.12 and 0.24. Which of these statements is a correct interpretation of the confidence level?

A
Approximately 95% of all animals in the shelter were adopted.
B
There is a 95% probability that the true proportion of all animals in the shelter that were adopted last month is between 0.12 and 0.24.
C
If many random samples of size 100 are selected from all records of animals in shelters, approximately 95% of the intervals would capture the true proportion that were adopted.
D
Approximately 95% of the sample proportions, based on random samples of size 100 from the population of all records of animals in shelters, will fall between 0.12 and 0.24.
4

Heather runs a successful lawn-mowing business. She would like to estimate the true mean amount of time it takes for her employees to mow a lawn. To do so, she selects a random sample of 30 customers and records the time it takes the employees to mow their lawns. The 90% confidence interval for the true mean time it takes for her employees to mow a lawn is 40–55 minutes. Which of the following would decrease the width of the interval?

A
selecting another sample
B
increasing the sample size
C
decreasing the sample size
D
increasing the confidence level
5

Heather runs a successful lawn-mowing business. She would like to estimate the true mean amount of time it takes for her employees to mow a lawn. To do so, she selects a random sample of 30 customers and records the time it takes the employees to mow their lawns. The 90% confidence interval for the true mean time it takes her employees to mow a lawn is 40 to 55 minutes. Which of these statements is a correct interpretation of the confidence level?

A
Approximately 90% of the customers have lawns that take between 40 and 55 minutes to mow.
B
There is a 90% probability that the true mean time it takes for her employees to mow a lawn is between 40 and 55 minutes.
C
If many random samples of size 30 are selected from the population of all customers, approximately 90% of the sample mean times for her employees to mow the lawns will be between 40 and 55 minutes.
D
If many random samples of size 30 are selected from the population of all customers, approximately 90% of the constructed intervals would capture the true mean time it takes for her employees to mow a lawn.
6

An animal rescue agent wanted to estimate the true proportion of all animals in shelters that are adopted each month. To do so, she selects a random sample of 100 animals that were in shelters last month and determines that the 95% confidence interval for the true proportion of animals adopted is between 0.12 and 0.24. This interval has a margin of error of 0.06. Which of the following can be accounted for by the margin of error?

A
response bias
B
nonresponse bias
C
sampling variation
D
undercoverage bias
7

An animal rescue agent wanted to estimate the true proportion of all animals in shelters that are adopted each month. To do so, she selects a random sample of 100 animals that were in shelters last month and determines that the 95% confidence interval for the true proportion of animals adopted each month is between 0.12 and 0.24. Which of the following would decrease the margin of error?

A
selecting another sample
B
increasing the sample size
C
decreasing the sample size
D
increasing the confidence level
8

A farmer of a large apple orchard would like to estimate the true mean number of suitable apples produced per tree. He selects a random sample of 40 trees from his large orchard and determines with 95% confidence that the true mean number of suitable apples produced per tree is between 375 and 520. Which of these statements is a correct interpretation of the confidence level?

A
Approximately 95% of the trees in the orchard contain between 375 and 520 suitable apples.
B
There is a 95% probability that the true mean number of apples per tree is between 375 and 520.
C
If many random samples of size 40 are selected from the population of all trees in this large orchard, approximately 95% of the sample means will be between 375 and 520 apples.
D
If many random samples of size 40 are selected from the population of all trees in this large orchard, about 95% of the intervals would capture the true mean number of suitable apples produced per tree.
9

A college professor would like to estimate the proportion of students who pull an "all-nighter,” meaning they study all night for an upcoming exam. She selects a random sample of 100 students from her large college and finds that the 99% confidence interval for the true proportion of students who have pulled an all-nighter to be 0.48 to 0.62. If the professor had randomly selected 50 students rather than 100 students, what effect would this have had on the width of the interval?

A
It would have doubled.
B
It would have been cut in half.
C
It would have remained the same.
D
It would have been larger, but it would not have doubled.
10

The dean of students at a large college is interested in learning about their opinions regarding the percentage of first-year students who should be given parking privileges in the main lot. He sends out an email survey to all students about this issue. A large number of first-year students reply but very few sophomores, juniors, and seniors reply. Based on the responses he receives, he constructs a 90% confidence interval for the true proportion of students who believe first-year students should be given parking privileges in the main lot to be (0.71, 0.79). Which of the following may have an impact on the confidence interval, but is not accounted for by the margin of error?

A
response bias
B
nonresponse bias
C
sampling variation
D
undercoverage bias

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