Exam 7: Inferences Based on a Single Sample: Estimation With Confidence Intervals

A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 98% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). What additional assumption is necessary for this confidence interval to be valid?

Sales of a new line of athletic footwear are crucial to the success of a company. The company wishes to estimate the average weekly sales of the new footwear to within $500 with 90% reliability. The initial sales indicate that the standard deviation of the weekly sales figures is approximately $1200. How many weeks of data must be sampled for the company to get the information it desires?

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight 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. Stummary statistics yield $\bar { x } = 180.975$ and $s = 143.042$ . Calculate a $99 \%$ confidence interval for the mean endowment of all private colleges in the United States. A) $180.975 \pm 176.955$ B) $180.975 \pm 189.173$ C) $180.975 \pm 169.672$ D) $180.975 \pm 181.387$ 181.387

We intend to estimate the average driving time of a group of commuters. From a previous study, we believe that the average time is 42 minutes with a standard deviation of 12 minutes. We want our 98 percent confidence interval to have a margin of error of no more than plus or minus 3 minutes. What is the smallest sample size that we should consider?

For n = 40 and $\hat { p }$ = .45, is the sample size large enough to construct a confidence for p?

The registrar's office at State University would like to determine a 95% confidence interval for the mean commute time of its students. A member of the staff randomly chooses a parking lot and surveys the first 125 students who park in the chosen lot on a given day. The confidence interval is

What type of car is more popular among college students, American or foreign? One hundred fifty -nine college students were randomly sampled and each was asked which type of car he or she prefers. A computer package was used to generate the printout below for the proportion of college students who prefer American automobiles. SAMPLE PROPORTION = .394141 SAMPLE SIZE = 159 UPPER LIMIT = .464240 LOWER LIMIT = .331153 What proportion of the sampled students prefer foreign automobiles?

A local men's clothing store is being sold. The buyers are trying to estimate the percentage of items that are outdated. They will choose a random sample from the 100,000 items in the store's inventory in order to determine the proportion of merchandise that is outdated. The current owners have never determined the percentage of outdated merchandise and cannot help the buyers. How large a sample do the buyers need in order to be 90% confident that the margin of error of their estimate is within 5%?

A marketing research company is estimating the average total compensation of CEOs in the service industry. Data were randomly collected from 18 CEOs and the 97% confidence interval for the mean was calculated to be ($2,181,260, $5,836,180). What would happen to the confidence interval if the confidence level were changed to 95%?

Suppose it is desired to estimate the average time a customer spends in a particular store to within 5 minutes (e.g., + 5 minutes) at 97% reliability. It is estimated that the standard deviation of the times is 15 minutes. How large a sample should be taken to get the desired interval?

A 90% confidence interval for the mean percentage of airline reservations being canceled on the day of the flight is (1.3%, 5%). What is the point estimator of the mean percentage of reservations that are canceled on the day of the flight?

What is the confidence level of the following confidence interval for ?? $\bar { x } \pm 1.96 \left( \frac { \sigma } { \sqrt { n } } \right)$

It is desired to estimate the average time it takes Statistics students to finish a computer project to within two hours at 97% reliability. It is estimated that the standard deviation of the times is 14 hours. How large a sample should be taken to get the desired interval?

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of eight private colleges in the United States revealed the following endowments (in millions of dollars): 64.5, 55.1, 232.8, 496.1, 127.6, 186.4, 104.7, and 212.2. What value will be used as the point estimate for the mean endowment of all private colleges in the United States?

What type of car is more popular among college students, American or foreign? One hundred fifty -nine college students were randomly sampled and each was asked which type of car he or she prefers.. A computer package was used to generate the printout below of a 90% confidence interval for the proportion of college students who prefer American automobiles. SAMPLE PROPORTION = .396 SAMPLE SIZE = 159 UPPER LIMIT = .472 LOWER LIMIT = .320 Which of the following is a correct practical interpretation of the interval?

An educator wanted to look at the study habits of university students. As part of the research, data was collected for three variables - the amount of time (in hours per week) spent studying, the amount of time (in hours per week) spent playing video games and the GPA - for a sample of 20 male university students. As part of the research, a 95% confidence interval for the average GPA of all male university students was calculated to be: (2.95, 3.10). The researcher claimed that the average GPA of all male students exceeded 2.94. Using the confidence interval supplied above, how do you respond to this claim?

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, $\bar { x } = 19.8 \text { and } s ^ { 2 } = 9$ . If the director wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 95% reliability, what is the minimum sample size she should use?

Which information is not shown on the screen below?

It is desired to estimate the proportion of college students who feel a sudden relief now that their statistics class is over. How many students must be sampled in order to estimate the true proportion to within 2% at the 90% confidence level?

Suppose you selected a random sample of n = 7 measurements from a normal distribution. Compare the standard normal z value with the corresponding t value for a 90% confidence interval.