Deck 8: Interval Estimation

Question 1

If we want to find the required sample size for the interval estimation of the population proportion, and no reasonable estimate of this proportion is available, we assume the worst-case scenario under which = 0.5.

Answer 1

True

Question 2

Excel does not have a special function that allows to calculate t values.

Answer 2

False

Question 3

A confidence interval is generally associated with a(n) ________.

Answer 3

margin of error

Question 4

For a given confidence level 100(1 - α)% and population standard deviation σ, the width of the confidence interval for the population mean is wider, the smaller the sample size n.

Question 5

The ________ determine the extent of the broadness of the tails of the distribution.

Question 6

The width of the confidence interval is ________ times the margin of error.

Question 7

To construct a 95% confidence interval for µ, the sampling distribution of must be ________.

Question 8

If a random sample of size n is taken from a normal population with a finite variance, then the statistic T = follows the t_df distribution with (n − 1) degrees of freedom, df.

Question 9

For a given confidence level 100(1 - α)% and sample size n, the width of the confidence interval for the population mean is narrower, the greater the population standard deviation σ.

Question 10

The t distribution table lists t_df values for selected lower-tail probabilities and degrees of freedom df.

Question 11

The sampling distribution of is based on a normal distribution on samples of any size.

Question 12

The t_df distribution has broader tails than the z distribution.

Question 13

The required sample size for the interval estimation of the population mean can be computed if we specify the population standard deviation σ, the value of z_α/2 based on the confidence level 100(1 − α)% and the desired margin of error D.

Question 14

Like the z distribution, the t_df distribution is symmetric around 0, bell-shaped, and with tails that approach the horizontal axis and eventually cross it.

Question 15

The t_df distribution consists of a family of distributions where the actual shape of each one depends on the degrees of freedom, df. For lower values of df, the t_df distribution is similar to the z distribution.

Question 16

The confidence coefficient is equal to ________.

Question 17

If a small segment of the population is sampled then an estimate will be less precise.

Question 18

The main ingredient for developing a confidence interval is the sampling distribution of the underlying statistic.

Question 19

For a given sample size n and population standard deviation σ, the width of the confidence interval for the population mean is wider, the smaller the confidence level 100(1 - α)%.

Question 20

A confidence interval provides a value that, with a certain measure of confidence, is the population parameter of interest.

Question 21

What is z_α/2 for a 95% confidence interval of the population mean?

A) 0.48
B) 0.49
C) 1.645
D) 1.96

Question 22

If calculated required sample size is a noninteger value, we should always ________ calculated value.

Question 23

At an academically challenging high school, the average GPA of a high school senior is known to be normally distributed with a variance of 0.25. A sample of 20 seniors is taken and their average GPA is found to be 2.71. Develop a 90% confidence interval for the population mean GPA.

A) [2.5261, 2.8939]
B) [2.5667, 2.8533]
C) [2.6180, 2.8020]
D) [2.6384, 2.7816]

Question 24

What is the most typical form of a calculated confidence interval?

A) Point estimate ± Standard error
B) Point estimate ± Margin of error
C) Population parameter ± Standard error
D) Population parameter ± Margin of error

Question 25

The daily revenue from the sale of fried dough at a local street vendor in Boston is known to be normally distributed with a known standard deviation of $120. The revenue on each of the last 25 days is noted, and the average is computed as $550. Construct a 95% confidence interval for the population mean of the sale of fried dough by this vendor.

A) 120 ± 1.645(550/5)
B) 120 ± 1.96(550/5)
C) 550 ± 1.645(120/5)
D) 550 ± 1.96(120/5)

Question 26

For a given confidence level and sample size, which of the following is true in the interval estimation of the population mean when σ is known?

A) If the sample standard deviation is smaller, the interval is wider.
B) If the population standard deviation is greater, the interval is wider.
C) If the population standard deviation is smaller, the interval is wider.
D) If the population standard deviation is greater, the interval is narrower.

Question 27

We draw a random sample of size 36 from a population with standard deviation 3.5. If the sample mean is 27, what is a 95% confidence interval for the population mean?

A) [25.8567, 28.1433]
B) [26.0405, 27.9595]
C) [26.8100, 27.1900]
D) [26.8401, 27.1599]

Question 28

An analyst takes a random sample of 25 firms in the telecommunications industry and constructs a confidence interval for the mean return for the prior year. Holding all else constant, if he increased the sample size to 30 firms, how are the standard error of the mean and the width of the confidence interval affected?

A) 1.645(3.20/6)
B) 1.645(10.24/6)
C) 1.96(3.20/6)
D) 1.96(10.24/6)

Question 29

The ages of MBA students at a university are normally distributed with a known population variance of 10.24. Suppose you are asked to construct a 95% confidence interval for the population mean age if the mean of a sample of 36 students is 26.5 years. If a 99% confidence interval is constructed instead of a 95% confidence interval for the population mean, then ________.

A) the resulting margin of error will increase and the risk of reporting an incorrect interval will increase
B) the resulting margin of error will decrease and the risk of reporting an incorrect interval will increase
C) the resulting margin of error will increase and the risk of reporting an incorrect interval will decrease
D) the resulting margin of error will decrease and the risk of reporting an incorrect interval will decrease

Question 30

We draw a random sample of size 25 from the normal population with variance 2.4. If the sample mean is 12.5, what is a 99% confidence interval for the population mean?

A) [11.2600, 13.7400]
B) [11.3835, 13.6165]
C) [11.7019, 13.2981]
D) [11.7793, 13.2207]

Question 31

For a given confidence level and population standard deviation, which of the following is true in the interval estimation of the population mean?

A) If the sample size is bigger, the interval is narrower.
B) If the sample size is smaller, the interval is narrower.
C) If the population size is bigger, the interval is narrower.
D) If the population size is smaller, the interval is narrower.

Question 32

The parameter p represents the ________ of successes in the population.

Question 33

The daily revenue from the sale of fried dough at a local street vendor in Boston is known to be normally distributed with a known standard deviation of $120. The revenue on each of the last 25 days is noted, and the average is computed as $550. A 95% confidence interval is constructed for the population mean revenue. If the data from the last 40 days had been used, then the resulting 95% confidence intervals would have been ________.

A) wider, with a larger probability of reporting an incorrect interval
B) wider, with the same probability of reporting an incorrect interval
C) narrower, with a larger probability of reporting an incorrect interval
D) narrower, with the same probability of reporting an incorrect interval

Question 34

It is important to know that ________ is increased when we estimate the population standard deviation with the sample standard deviation.

Question 35

An analyst takes a random sample of 25 firms in the telecommunications industry and constructs a confidence interval for the mean return for the prior year. Holding all else constant, if he increased the sample size to 30 firms, how are the standard error of the mean and the width of the confidence interval affected?

A) A
B) B
C) C
D) D

Question 36

What is the purpose of calculating a confidence interval?

A) To provide a range of values that has a certain large probability of containing the sample statistic of interest.
B) To provide a range of values that, with a certain measure of confidence, contains the sample statistic of interest.
C) To provide a range of values that, with a certain measure of confidence, contains the population parameter of interest.
D) To provide a range of values that has a certain large probability of containing the population parameter of interest.

Question 37

A sample of a given size is used to construct a 95% confidence interval for the population mean with a known population standard deviation. If a bigger sample had been used instead, then the 95% confidence interval would have been ________ and the probability of making an error would have been ________.

A) wider; smaller
B) narrower; smaller
C) wider; unchanged
D) narrower; unchanged

Question 38

Which of the following is the necessary condition for creating confidence intervals for the population mean?

A) Normality of the estimator
B) Normality of the population
C) Known population parameter
D) Known standard deviation of the estimator

Question 39

What is z_α/2 for a 90% confidence interval of the population mean?

A) 0.48
B) 0.49
C) 1.645
D) 1.96

Question 40

In an examination of purchasing patterns of shoppers, a sample of 16 shoppers revealed that they spent, on average, $54 per hour of shopping. Based on previous years, the population standard deviation is thought to be $21 per hour of shopping. Assuming that the amount spent per hour of shopping is normally distributed, find a 90% confidence interval for the mean amount.

A) [$45.36, $62.64]
B) [$47.27, $60.73]
C) [$51.84, $56.16]
D) [$52.32, $55.68]

Question 41

The GPA of accounting students in a university is known to be normally distributed. A random sample of 20 accounting students results in a mean of 2.92 and a standard deviation of 0.16. Construct the 95% confidence interval for the mean GPA of all accounting students at this university.

A) 2.92 ± 1.729(0.16/ )
B) 2.92 ± 1.96(0.16/ )
C) 2.92 ± 2.086(0.16/ )
D) 2.92 ± 2.093(0.16/ )

Question 42

The mortgage foreclosure crisis that preceded the Great Recession impacted the U.S. economy in many ways, but it also impacted the foreclosure process itself as community activists better learned how to delay foreclosure, and lenders became more wary of filing faulty documentation. Suppose the duration of the eight most recent foreclosures filed in the city of Boston (from the beginning of foreclosure proceedings to the filing of the foreclosure deed, transferring the property) has been 230 days, 420 days, 340 days, 367 days, 295 days, 314 days, 385 days, and 311 days. Assume the duration is normally distributed. Construct a 99% confidence interval for the mean duration of the foreclosure process in Boston.

A) [126.2730, 539.2270]
B) [259.7393, 405.7607]
C) [279.0056, 386.4944]
D) [291.8600, 373.6400]

Question 43

In an examination of holiday spending (known to be normally distributed) of a sample of 16 holiday shoppers at a local mall, an average of $54 was spent per hour of shopping. Based on the current sample, the standard deviation is equal to $21. Find a 90% confidence interval for the population mean level of spending per hour.

A) [$44.83, $63.17]
B) [$44.79, $63.20]
C) [$45.36, $62.63]
D) [$46.96, $61.04]

Question 44

A company that produces computers recently tested the battery in its latest laptop in six separate trials. The battery lasted 8.23,7.89,8.14,8.25,8.30, and 7.95 hours before burning out in each of the tests. Assuming the battery duration is normally distributed, construct a 95% confidence interval for the mean battery life in the new model.

A) [7.9490, 8.3044]
B) [7.9575, 8.2959]
C) [7.9873, 8.2661]
D) [7.9912, 8.2622]

Question 45

At a particular academically challenging high school, the average GPA of a high school senior is known to be normally distributed. After a sample of 20 seniors is taken, the average GPA is found to be 2.71 and the variance is determined to be 0.25. Find a 90% confidence interval for the population mean GPA.

A) [2.5167, 2.9033]
B) [2.5261, 2.8961]
C) [2.5615, 2.8585]
D) [2.6358, 2.7842]

Question 46

Suppose taxi fares from Logan Airport to downtown Boston is known to be normally distributed and a sample of seven taxi fares produces a mean fare of $22.31 and a 95% confidence interval of [$20.5051, $24.2091]. Which of the following statements is a valid explanation of the confidence interval.

A) 95% of all taxi fares are between $20.51 and $24.21.
B) We are 95% confident that a randomly selected taxi fare will be between $20.51 and $24.21.
C) The mean amount of a taxi fare is $22.31, 95% of the time.
D) We are 95% confident that the average taxi fare between Logan Airport and downtown Boston will fall between $20.51 and $24.21.

Question 47

The starting salary of business students in a university is known to be normally distributed. A random sample of 18 business students results in a mean salary of $46,500 with a standard deviation of $10,200. Construct the 90% confidence interval for the mean starting salary of business students in this university.

A) 46,500 ± 1.330(10,200/ )
B) 46,500 ± 1.645(10,200/ )
C) 46,500 ± 1.734(10,200/ )
D) 46,500 ± 1.740(10,200/ )

Question 48

How do the t_df and z distributions differ?

A) There is no difference between the t_df and z distributions; they both do not have asymptomatic tails.
B) The z distribution has broader tails (it is flatter around zero).
C) The t_df distribution has broader tails (it is flatter around zero).
D) The z distribution has asymptotic tails, while the t_df distribution does not.

Question 49

Given a sample mean of 12.5-drawn from a normal population, a sample of size 25, and a sample variance of 2.4-find a 99% confidence interval for the population mean.

A) [11.7019, 13.2981]
B) [11.1574, 13.8426]
C) [11.6334, 13.3666]
D) [11.7279, 13.2721]

Question 50

Suppose taxi fare from Logan Airport to downtown Boston is known to be normally distributed with a standard deviation of $2.50. The last seven times John has taken a taxi from Logan to downtown Boston, the fares have been $22.10, $23.25, $21.35, $24.50, $21.90, $20.75, and $22.65. What is a 95% confidence interval for the population mean taxi fare?

A) [$17.46, $27.26]
B) [$20.46, $24.16]
C) [$20.80, $23.91]
D) [$21.66, $23.06]

Question 51

What is t_α/2,df for a 99% confidence interval of the population mean based on a sample of 25 observations?

A) 2.492
B) 2.576
C) 2.787
D) 2.797

Question 52

Confidence intervals of the population mean may be created for the cases when the population standard deviation is known or unknown. How are these two cases treated differently?

A) Use the z table when s is unknown; use the t table when s is known.
B) Use the z table when s is known; use the t table when s is unknown.
C) Use the z table when σ is unknown; use the t table when σ is known.
D) Use the z table when σ is known; use the t table when σ is unknown.

Question 53

What conditions are required by the central limit theorem before a confidence interval of the population mean may be created?

A) The underlying population must be normally distributed.
B) The underlying population must be normally distributed if the sample size is 30 or more.
C) The underlying population need not be normally distributed if the sample size is 30 or more.
D) The underlying population need not be normally distributed if the population standard deviation is known.

Question 54

A 90% confidence interval is constructed for the population mean. If a 95% confidence interval had been constructed instead (everything else remaining the same), the width of the interval would have been ________ and the probability of making an error would have been ________.

A) wider; bigger
B) wider; smaller
C) narrower; bigger
D) narrower; smaller

Question 55

For a given sample size and population standard deviation, which of the following is true in the interval estimation of the population mean?

A) If the confidence level is greater, the interval is wider.
B) If the confidence level is smaller, the interval is wider.
C) If the confidence level is greater, the interval is narrower.
D) None of the above

Question 56

What is t_α/2,df for a 95% confidence interval of the population mean based on a sample of 15 observations?

A) 1.761
B) 1.960
C) 2.131
D) 2.145

Question 57

The mortgage foreclosure crisis that preceded the Great Recession impacted the U.S. economy in many ways, but it also impacted the foreclosure process itself as community activists better learned how to delay foreclosure and lenders became more wary of filing faulty documentation. Suppose the duration of the eight most recent foreclosures filed in the city of Boston (from the beginning of foreclosure proceedings to the filing of the foreclosure deed, transferring the property) has been 230 days, 420 days, 340 days, 367 days, 295 days, 314 days, 385 days, and 311 days. Assume the duration is normally distributed. Construct a 90% confidence interval for the mean duration of the foreclosure process in Boston.

A) [259.7400, 405.7600]
B) [293.2229, 372.2771]
C) [298.4296, 367.0704]
D) [303.2282, 362.2718]

Question 58

A website advertises job openings on its website, but job seekers have to pay to access the list of job openings. The website recently completed a survey to estimate the number of days it takes to find a new job using its service. It took the last 30 customers an average of 60 days to find a job. Assume the population standard deviation is 10 days. Calculate a 95% confidence interval of the population mean number of days it takes to find a job.

A) [40.4000, 79.6000]
B) [55.9085, 64.0915]
C) [56.4216, 63.5784]
D) [56.9966, 63.0034]

Question 59

The t_df distribution is similar to the z distribution because ________.

A) as the degrees of freedom go to infinity, the t distribution converges to the z distribution, but both do not have asymptomatic tails
B) both have asymptotic tails-that is, their tails become closer and closer to the horizontal axis and eventually cross the axis
C) as the degrees of freedom go to infinity, the t distribution converges to the z distribution and both have asymptotic tails-that is, their tails become closer and closer to the horizontal axis but never touch it
D) Neither as the degrees of freedom go to infinity, the t distribution converges to the z distribution nor both have asymptotic tails-that is, their tails become closer and closer to the horizontal axis but never touch it

Question 60

Given a sample mean of 27 and a sample standard deviation of 3.5 computed from a sample of size 36, find a 95% confidence interval on the population mean.

A) [25.8158, 28.1842]
B) [25.8170, 28.1830]
C) [26.0142, 27.9858]
D) [26.9930, 27.0070]

Question 61

The sampling distribution of the population proportion is based on a binomial distribution. What condition must be met to use the normal approximation for the confidence interval?

A) n ≥ 30
B) n ≥ 5
C) n ≥ 5 and n(1 - ) ≥ 5
D) np ≥ 5 and n(1 - p) ≥ 5

Question 62

A sample of 1,400 American households was asked if they planned to buy a new car next year. Of the respondents, 34% indicated they planned to buy a new car next year. Construct a 98% confidence interval of the proportion of American households who expect to buy a new car next year.

A) [0.3074, 0.3726]
B) [0.3105, 0.3695]
C) [0.3140, 0.3660]
D) [0.3392, 0.3408]

Question 63

A basketball coach wants to know how many free throws an NBA player shoots during the course of an average practice. The coach takes a random sample of 43 players and finds the average number of free throws shot per practice was 225 with a standard deviation of 35. Construct a 99% confidence interval for the average number of free throws in practice.

A) [210.5992, 239.4008]
B) [210.6155, 239.3845]
C) [211.2506, 238.7494]
D) [214.2290, 235.7710]

Question 64

A sample of 1,400 American households was asked if they planned to buy a new car next year. Of the respondents, 34% indicated they planned to buy a new car next year. Construct a 90% confidence interval of the proportion of American households who expect to buy a new car next year.

A) [0.3152, 0.3648]
B) [0.3192, 0.3608]
C) [0.3394, 0.3406]
D) [0.3354, 0.3446]

Question 65

When the required sample size calculated by using a formula is not a whole number, what is the best choice for the required sample size?

A) Round the result of the calculation in a standard way.
B) Round the result of the calculation up to the nearest whole number.
C) Round the result of the calculation down to the nearest whole number.
D) None of these choices are correct.

Question 66

Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, 98 supported candidate B and 58 supported candidate C. Construct a 99% confidence interval on the population proportion for the support of candidate A in the following election.

A) [0.4057, 0.5543]
B) [0.4130, 0.5470]
C) [0.4779, 0.4821]
D) [0.4781, 0.4819]

Question 67

Professors of accountancy are in high demand at American universities. A random sample of 28 new accounting professors found the average salary was $135 thousand with a standard deviation of $16 thousand. Assume the distribution is normally distributed. Construct a 95% confidence interval for the salary of new accounting professors. Answers are in thousands of dollars.

A) [102.1680, 167.832]
B) [127.8247, 142.1753]
C) [128.7958, 141.2042]
D) [129.0735, 140.9265]

Question 68

According to a report in USA Today, more and more parents are helping their young adult children get homes. Suppose eight persons in a random sample of 40 young adults who recently purchased a home in Kentucky received help from their parents. You have been asked to construct a 95% confidence interval for the population proportion of all young adults in Kentucky who received help from their parents. If a 99% confidence interval is constructed instead of a 95% confidence interval for the population proportion, then ________.

A) the resulting margin of error will increase and the risk of reporting an incorrect interval will increase
B) the resulting margin of error will decrease and the risk of reporting an incorrect interval will increase
C) the resulting margin of error will increase and the risk of reporting an incorrect interval will decrease
D) the resulting margin of error will decrease and the risk of reporting an incorrect interval will decrease

Question 69

According to a report in USA Today, more and more parents are helping their young adult children get homes. Suppose eight persons in a random sample of 40 young adults who recently purchased a home in Kentucky received help from their parents. You have been asked to construct a 95% confidence interval for the population proportion of all young adults in Kentucky who received help from their parents. What is the margin of error for a 95% confidence interval for the population proportion?

A) 1.645(0.0040)
B) 1.645(0.0632)
C) 1.96(0.0040)
D) 1.96(0.0632)

Question 70

The confidence intervals for the population proportion are generally based on ________.

A) the z distribution
B) the t distribution
C) the z distribution when the sample size is very small
D) the t distribution when the population standard deviation is not known

Question 71

When examining the possible outcome of an election, what type of confidence interval is most suitable for estimating the current support for a candidate?

A) The confidence interval for the sample mean
B) The confidence interval for the population mean
C) The confidence interval for the sample proportion
D) The confidence interval for the population proportion

Question 72

A sample of 2,007 American adults was asked how they viewed China, with 17% of respondents calling the country "unfriendly" and 6% of respondents indicating the country was "an enemy." Construct a 99% confidence interval of the proportion of American adults who viewed China as "an enemy."

A) [0.0463, 0.0737]
B) [0.0476, 0.0724]
C) [0.0512, 0.0688]
D) [0.0597, 0.0603]

Question 73

Professors of accountancy are in high demand at American universities. A random sample of 28 new accounting professors found the average salary was $135 thousand with a standard deviation of $16 thousand. Assume the distribution is normally distributed. Construct a 90% confidence interval for the salary of new accounting professors. Answers are in thousands of dollars.

A) [107.7520, 162.2480]
B) [129.8497, 140.1503]
C) [130.0260, 139.9740]
D) [131.0268, 138.9732]

Question 74

What is the minimum sample size required to estimate a population mean with 90% confidence when the desired margin of error is D = 1.25? The standard deviation in a preselected sample is 8.5.

A) n = 76
B) n = 125
C) n = 126
D) n = 190

Question 75

Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, 153 supported her. Construct a 90% confidence interval on the population proportion for the support of candidate A in the following election.

A) [0.4534, 0.5666]
B) [0.4625, 0.5575]
C) [0.5086, 0.5114]
D) [0.5090, 0.5110]

Question 76

Candidate A is facing two opposing candidates in a mayoral election. In a recent poll of 300 residents, she has garnered 51% support. Construct a 95% confidence interval on the population proportion for the support of candidate A in the following election.

A) [0.4534, 0.5666]
B) [0.4625, 0.5575]
C) [0.5084, 0.5116]
D) [0.5086, 0.5114]

Question 77

Which of the following is the correct formula for the margin of error in the interval estimation of p?

A) z_α/2
B) z_α/2
C) z_α/2
D) z_α/2

Question 78

What is the minimum sample size required to estimate a population mean with 95% confidence when the desired margin of error is E = 1.5? The population standard deviation is known to be 10.75.

A) n = 138
B) n = 139
C) n = 197
D) n = 198

Question 79

A sample of 2,007 American adults was asked how they viewed China, with 17% of respondents calling the country "unfriendly" and 6% of respondents indicating the country was "an enemy." Construct a 95% confidence interval of the proportion of American adults who viewed China as either "unfriendly" or "an enemy."

A) [0.2013, 0.2587]
B) [0.2116, 0.2484]
C) [0.2146, 0.2454]
D) [0.2208, 0.2392]

Question 80

Statisticians like precision in their interval estimates. A low margin of error is needed to achieve this. Which of the following supports this when selecting sample sizes?

A) A larger sample size reduces the margin of error.
B) A smaller sample size reduces the margin of error.
C) A larger sample size increases the margin of error.
D) A sample size has no impact on the margin of error.