Ch. 8 Stats

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 ________.

narrower; unchanged

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

Normality of the estimator

How do the tdf and z distributions differ?

The tdf distribution has broader tails (it is flatter around zero).

What is the purpose of calculating a confidence interval?

To provide a range of values that, with a certain measure of confidence, contains the population parameter of interest.

If a random sample of size n is taken from a normal population with a finite variance, then the statistic T = /X-u/S sqrt(n) follows the tdf distribution with (n-1) degrees of freedom, df.

True

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

True

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

Use the z table when σ is known; use the t table when σ is unknown

Professors of accountancy are in high demand at American universities. A random 36) 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.

[128.7958, 141.2042]

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.

[25.8158, 28.1842]

The tdf distribution is similar to the z distribution because ___________________________________________.

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

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

1.96

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

False

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.

False

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)%.

False

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.

False

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

2.145

The 𝑡 (𝑑𝑓) distribution has broader tails than the z distribution.

True

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.

2.92 ± 2.093(0.16/sqrt(20))

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.

550 ± 1.96(120/5)

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 Increases Becomes wider B Increases Becomes narrower C Decreases Becomes wider D Decreases Becomes narrower

D

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

If the confidence level is greater, the interval is wider.

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?

If the population standard deviation is greater, the interval is wider.

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

Point estimate ± Margin of error

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.

True

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.

[$44.79, $63.20]

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.

[$45.36, $62.64]

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.

[0.4057, 0.5543]

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.

[0.4534, 0.5666]

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.

[0.4625, 0.5575]

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.

[259.7393, 405.7607]

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 _______________________________________________.

the resulting margin of error will increase and the risk of reporting an incorrect interval will decrease

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

the z distribution

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

za/2 sqrt /p(1-/p)/n