Deck 8: Confidence Interval Estimation

Question 1

For a given data set, the confidence interval will be wider for 95% confidence than for 90% confidence.

Answer 1

True

Question 2

The confidence interval obtained will always correctly estimate the population parameter.

Answer 2

False

Question 3

The sampling error can either be positive or negative.

Answer 3

True

Question 4

A population parameter is used to estimate a confidence interval.

Question 5

Suppose a 95% confidence interval for μ turns out to be (1,000, 2,100). To make more useful inferences from the data, it is desired to reduce the width of the confidence interval. Which of the following will result in a reduced interval width?

A) Increase the sample size
B) Increase the confidence level
C) Increase the population mean
D) Increase the sample mean

Question 6

The difference between the lower limit of a confidence interval and the point estimate used in constructing the confidence interval is called the sampling error.

Question 7

Holding the level of confidence fixed, increasing the sample size will lead to a wider confidence interval.

Question 8

Holding the width of a confidence interval fixed, increasing the level of confidence can be achieved with a lower sample size.

Question 9

Suppose a 95% confidence interval for μ has been constructed. If it is decided to take a larger sample and to decrease the confidence level of the interval, then the resulting interval width would ________. (Assume that the sample statistics gathered would not change very much for the new sample.)

A) be larger than the current interval width
B) be narrower than the current interval width
C) be the same as the current interval width
D) be unknown until actual sample sizes and reliability levels were determined

Question 10

In the construction of confidence intervals, if all other quantities are unchanged, an increase in the sample size will lead to a ________ interval.

A) narrower
B) wider
C) less significant
D) biased

Question 11

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. What value will be used as the point estimate for the mean endowment of all private colleges in the United States?

A) $1,447.8
B) $180.975
C) $143.042
D) $8

Question 12

The difference between the upper limit of a confidence interval and the point estimate used in constructing the confidence interval is called the sampling error.

Question 13

A point estimate consists of a single sample statistic that is used to estimate the true population parameter.

Question 14

The sample mean is a point estimate of the population mean.

Question 15

Sampling error equals half the width of a confidence interval.

Question 16

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. What is an efficient, unbiased point estimate of the number of books checked out each day at the Library of Congress?

A) 740
B) 830
C) 920
D) 1,660

Question 17

Holding the sample size fixed, increasing the level of confidence in a confidence interval will necessarily lead to a wider confidence interval.

Question 18

Other things being equal, as the confidence level for a confidence interval increases, the width of the interval increases.

Question 19

The width of a confidence interval equals twice the sampling error.

Question 20

Other things being equal, the confidence interval for the mean will be wider for 95% confidence than for 90% confidence.

Question 21

The t distribution is used to construct confidence intervals for the population mean when the population standard deviation is unknown.

Question 22

An economist is interested in studying the incomes of consumers in a particular country. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in a mean income of $15,000. What is the upper end point in a 99% confidence interval for the average income?

A) $15,052
B) $15,141
C) $15,330
D) $15,364

Question 23

Which of the following is not true about the Student's t distribution?

A) It has more area in the tails and less in the center than does the normal distribution.
B) It is used to construct confidence intervals for the population mean when the population standard deviation is known.
C) It is bell shaped and symmetrical.
D) As the number of degrees of freedom increases, the t distribution approaches the normal distribution.

Question 24

Suppose a 95% confidence interval for μ turns out to be (1,000, 2,100). Give a definition of what it means to be "95% confident" as an inference.

A) In repeated sampling, the population parameter would fall in the given interval 95% of the time.
B) In repeated sampling, 95% of the intervals constructed would contain the population mean.
C) 95% of the observations in the entire population fall in the given interval.
D) 95% of the observations in the sample fall in the given interval.

Question 25

The difference between the sample proportion and the population proportion is called the sampling error.

Question 26

If you were constructing a 99% confidence interval of the population mean based on a sample of n = 25 where the standard deviation of the sample S = 0.05, the critical value of t will be

A) 2.7969.
B) 2.7874.
C) 2.4922.
D) 2.4851.

Question 27

The difference between the sample mean and the population mean is called the sampling error.

Question 28

The t distribution is used to develop a confidence interval estimate of the population mean when the population standard deviation is unknown.

Question 29

Given a sample mean of 2.1 and a population standard deviation of 0.7 from a sample of 10 data points, a 90% confidence interval will have a width of 2.36.

Question 30

For a t distribution with 12 degrees of freedom, the area between -2.6810 and 2.1788 is 0.980.

Question 31

It is desired to estimate the mean total compensation of CEOs in the Service industry. Data were randomly collected from 18 CEOs and the 95% confidence interval was calculated to be ($2,181,260, $5,836,180). Which of the following interpretations is correct?

A) 95% of the sampled total compensation values fell between $2,181,260 and $5,836,180.
B) We are 95% confident that the mean of the sampled CEOs falls in the interval $2,181,260 to $5,836,180.
C) In the population of Service industry CEOs, 95% of them will have total compensations that fall in the interval $2,181,260 to $5,836,180.
D) We are 95% confident that the mean total compensation of all CEOs in the Service industry falls in the interval $2,181,260 to $5,836,180.

Question 32

An economist is interested in studying the incomes of consumers in a particular country. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in a mean income of $15,000. What is the width of the 90% confidence interval?

A) $232.60
B) $364.30
C) $465.23
D) $728.60

Question 33

It is desired to estimate the mean total compensation of CEOs in the Service industry. Data were randomly collected from 18 CEOs and the 95% confidence interval was calculated to be ($2,181,260, $5,836,180). Based on the interval above, do you believe the mean total compensation of CEOs in the Service industry is more than $3,000,000?

A) Yes, and I am 95% confident of it.
B) Yes, and I am 78% confident of it.
C) I am 95% confident that the mean compensation is $3,000,000.
D) I cannot conclude that the mean exceeds $3,000,000 at the 95% confidence level.

Question 34

A 99% confidence interval estimate can be interpreted to mean that

A) if all possible samples of size n are taken and confidence interval estimates are developed, 99% of them would include the true population mean somewhere within their interval.
B) we have 99% confidence that we have selected a sample whose interval does include the population mean.
C) Both of the above.
D) None of the above.

Question 35

The t distribution

A) assumes the population is normally distributed.
B) approaches the normal distribution as the sample size increases.
C) has more area in the tails than does the normal distribution.
D) All of the above.

Question 36

A sample size of 5 provides a sample mean of 9.6. If the population variance is known to be 5 and the population distribution is assumed to be normal, the lower limit for a 90% confidence interval is 7.96.

Question 37

The difference between the sample size and the population size is called the sampling error.

Question 38

A major department store chain is interested in estimating the mean amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: X = $50.50 and S = 20. Assuming the distribution of the amount spent on their first visit is approximately normal, what is the shape of the sampling distribution of the sample mean that will be used to create the desired confidence interval for μ?

A) Approximately normal with a mean of $50.50
B) A standard normal distribution
C) A t distribution with 15 degrees of freedom
D) A t distribution with 14 degrees of freedom

Question 39

The confidence interval estimate of the population mean is constructed around the sample mean.

Question 40

The t distribution approaches the standardized normal distribution when the number of degrees of freedom increases.

Question 41

A prison official wants to estimate the proportion of cases of recidivism. Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism. A 99% confidence interval for the proportion of cases of recidivism would go from ________ to ________.

Question 42

The t distribution allows the calculation of confidence intervals for means for small samples when the population variance is not known, regardless of the shape of the distribution in the population.

Question 43

The confidence interval estimate of the population proportion is constructed around the sample proportion.

Question 44

A major department store chain is interested in estimating the mean amount its credit card customers spent on their first visit to the chain's new store in the mall. Fifteen credit card accounts were randomly sampled and analyzed with the following results: X = $50.50 and S = 20. Construct a 95% confidence interval for the mean amount its credit card customers spent on their first visit to the chain's new store in the mall assuming that the amount spent follows a normal distribution.

A) $50.50 ± $9.09
B) $50.50 ± $10.12
C) $50.50 ± $11.00
D) $50.50 ± $11.08

Question 45

The county clerk wants to estimate the proportion of voters who will need special election facilities. Suppose a sample of 400 voters was taken. If 150 need special election facilities, calculate a 90% confidence interval for the population proportion.

Question 46

Given a sample mean of 2.1 and a sample standard deviation of 0.7 from a sample of 10 data points, a 90% confidence interval will have a width of 2.36.

Question 47

A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. Use a 90% confidence interval to estimate the true proportion of students who receive financial aid.

Question 48

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is put into a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 8 private colleges in the United States revealed the following endowments (in millions of dollars): 60.2, 47.0, 235.1, 490.0, 122.6, 177.5, 95.4, and 220.0. Summary statistics yield X = 180.975 and S = 143.042 . Calculate a 95% confidence interval for the mean endowment of all the private colleges in the United States assuming a normal distribution for the endowments.

A) $180.975 ± $94.066
B) $180.975 ± $99.123
C) $180.975 ± $116.621
D) $180.975 ± $119.586

Question 49

In estimating the population mean with the population standard deviation unknown, if the sample size is 12, there will be 6 degrees of freedom.

Question 50

A random sample of 50 provides a sample mean of 31 with a standard deviation of S = 14. The upper bound of a 90% confidence interval estimate of the population mean is 34.32.

Question 51

The county clerk wants to estimate the proportion of voters who will need special election facilities. The clerk wants to construct a 95% confidence interval for the population proportion which extends at most 0.07 to either side of the sample proportion. How large a sample must be taken to assure these conditions are met?

Question 52

As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, X = 19.8 and S = 5. Which of the following assumptions is necessary in order for a confidence interval to be valid?

A) The population sampled from has an approximate normal distribution.
B) The population sampled from has an approximate t distribution.
C) The mean of the sample equals the mean of the population.
D) None of these assumptions are necessary.

Question 53

As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, X = 19.8 and S² = 25. Estimate the mean number of admissions per 24-hour period with a 95% confidence interval.

Question 54

The t distribution allows the calculation of confidence intervals for means when the actual standard deviation is not known.

Question 55

A race car driver tested his car for time from 0 to 60 mph, and for 20 tests obtained a mean of 4.85 seconds with a standard deviation of 1.47 seconds. A 95% confidence interval for the 0 to 60 mean time is 4.52 seconds to 5.18 seconds.

Question 56

In forming a 90% confidence interval for a population mean from a sample size of 22, the number of degrees of freedom from the t distribution equals 22.

Question 57

A prison official wants to estimate the proportion of cases of recidivism. Examining the records of 250 convicts, the official determines that there are 65 cases of recidivism. A confidence interval will be obtained for the proportion of cases of recidivism. Part of this calculation includes the estimated standard error of the sample proportion. For this sample, the estimated standard error is ________.

Question 58

A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. The 95% confidence interval for π is 0.59 ± 0.07. Interpret this interval.

A) We are 95% confident that the true proportion of all students receiving financial aid is between 0.52 and 0.66.
B) 95% of the students get between 52% and 66% of their tuition paid for by financial aid.
C) We are 95% confident that between 52% and 66% of the sampled students receive some sort of financial aid.
D) We are 95% confident that 59% of the students are on some sort of financial aid.

Question 59

A quality control engineer is interested in estimating the proportion of defective items coming off a production line. In a sample of 300 items, 27 are defective. A 90% confidence interval for the proportion of defectives from this production line would go from ________ to ________.

Question 60

A sample of 100 fuses from a very large shipment is found to have 10 that are defective. The 95% confidence interval would indicate that, for this shipment, the proportion of defective fuses is between 0 and 0.28.

Question 61

TABLE 8-1
The managers of a company are worried about the morale of their employees. In order to determine if a problem in this area exists, they decide to evaluate the attitudes of their employees with a standardized test. They select the Fortunato test of job satisfaction, which has a known standard deviation of 24 points.
Referring to Table 8-1, due to financial limitations, the managers decide to take a sample of 45 employees. This yields a mean score of 88.0 points. A 90% confidence interval would go from ________ to ________.

Question 62

An economist is interested in studying the incomes of consumers in a particular country. The population standard deviation is known to be $1,000. A random sample of 50 individuals resulted in a mean income of $15,000. What total sample size would the economist need to use for a 95% confidence interval if the width of the interval should not be more than $100?

A) n = 1537
B) n = 385
C) n = 40
D) n = 20

Question 63

The width of a confidence interval estimate for a proportion will be

A) narrower for 99% confidence than for 95% confidence.
B) wider for a sample size of 100 than for a sample size of 50.
C) narrower for 90% confidence than for 95% confidence.
D) narrower when the sample proportion is 0.50 than when the sample proportion is 0.20.

Question 64

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, approximately how large a sample did her assistant use to determine the interval estimate?

A) 2
B) 3
C) 12
D) It cannot be determined from the information given.

Question 65

As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, X = 19.8 and S² = 25. Using the sample standard deviation as an estimate for the population standard deviation, what size sample should the director choose if she wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 99% reliability?

Question 66

When determining the sample size for a proportion for a given level of confidence and sampling error, the closer to 0.50 that π is estimated to be, the sample size required ________.

A) is smaller
B) is larger
C) is not affected
D) can be smaller, larger or unaffected

Question 67

A confidence interval was used to estimate the proportion of statistics students who are females. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Based on the interval above, is the population proportion of females equal to 0.60?

A) No, and we are 90% sure of it.
B) No. The proportion is 54.17%.
C) Maybe. 0.60 is a believable value of the population proportion based on the information above.
D) Yes, and we are 90% sure of it.

Question 68

The standardized normal distribution is used to develop a confidence interval estimate of the population proportion regardless of whether the population standard deviation is known.

Question 69

Suppose a department store wants to estimate the mean age of the customers of its contemporary apparel department, correct to within 2 years, with level of confidence equal to 95%. Management believes that the standard deviation is 8 years. The sample size they should take is ________.

Question 70

The standardized normal distribution is used to develop a confidence interval estimate of the population proportion when the sample size is sufficiently large.

Question 71

When determining the sample size necessary for estimating the true population mean, which factor is not considered when sampling with replacement?

A) The population size
B) The population standard deviation
C) The level of confidence desired in the estimate
D) The allowable or tolerable sampling error

Question 72

A university system enrolling hundreds of thousands of students is considering a change in the way students pay for their education. Currently, the students pay $400 per credit hour. The university system administrators are contemplating charging each student a set fee of $7,000 per quarter, regardless of how many credit hours each takes. To see if this proposal would be economically feasible, the administrators would like to know how many credit hours, on the average, each student takes per quarter. A random sample of 250 students yields a mean of 14.1 credit hours per quarter and a standard deviation of 2.3 credit hours per quarter. Suppose the administration wanted to estimate the mean to within 0.1 hours at 95% reliability and assumed that the sample standard deviation provided a good estimate for the population standard deviation. How large a total sample would they need to take?

Question 73

TABLE 8-1
The managers of a company are worried about the morale of their employees. In order to determine if a problem in this area exists, they decide to evaluate the attitudes of their employees with a standardized test. They select the Fortunato test of job satisfaction, which has a known standard deviation of 24 points.
Referring to Table 8-1, this confidence interval is only valid if the scores on the Fortunato test are normally distributed.

Question 74

A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 99% reliability, how many students would need to be sampled?

A) n = 1,844
B) n = 1,784
C) n = 1,503
D) n = 1,435

Question 75

TABLE 8-1
The managers of a company are worried about the morale of their employees. In order to determine if a problem in this area exists, they decide to evaluate the attitudes of their employees with a standardized test. They select the Fortunato test of job satisfaction, which has a known standard deviation of 24 points.
Referring to Table 8-1, they should sample ________ employees if they want to estimate the mean score of the employees within 5 points with 90% confidence.

Question 76

A confidence interval was used to estimate the proportion of statistics students who are female. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Using the information above, what total size sample would be necessary if we wanted to estimate the true proportion to within ±0.08 using 95% confidence?

A) 105
B) 150
C) 420
D) 597

Question 77

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, and she asked her assistant to use 25 days of data to construct the interval estimate, what confidence level can she attach to the interval estimate?

A) 99.7%
B) 99.0%
C) 98.0%
D) 95.4%

Question 78

TABLE 8-2
A quality control engineer is interested in the mean length of sheet insulation being cut automatically by machine. The desired mean length of the insulation is 12 feet. It is known that the standard deviation in the cutting length is 0.15 feet. A sample of 70 cut sheets yields a mean length of 12.14 feet. This sample will be used to obtain a 99% confidence interval for the mean length cut by machine.
Referring to Table 8-2, the critical value to use in obtaining the confidence interval is ________.

Question 79

The t distribution is used to develop a confidence interval estimate of the population proportion when the population standard deviation is unknown.

Question 80

The head librarian at the Library of Congress has asked her assistant for an interval estimate of the mean number of books checked out each day. The assistant provides the following interval estimate: from 740 to 920 books per day. If the head librarian knows that the population standard deviation is 150 books checked out per day, and she asked her assistant for a 95% confidence interval, approximately how large a sample did her assistant use to determine the interval estimate?

A) 125
B) 13
C) 11
D) 4