10.3 Warm Up (#2)

A random sample of 10 employees of a company was selected to estimate the mean one-way commute time for all employees at the company. The mean and standard deviation of the sample were 38 minutes and 6 minutes, respectively. Assuming all conditions for inference are met, which of the following is the margin of error, in minutes, for a 95 percent confidence interval for the population mean one-way commute time?

Answer explanation

Independent random samples of voters from two voting districts, G and H, were selected to investigate the proportion of all voters who favor a proposal to widen a road that runs through both districts. The difference between the sample proportions (G minus H) was used to create the 95 percent confidence interval (0.13, 0.47) for the population difference between districts. Which of the following is the best interpretation of the interval?

• It is likely that the majority of voters in both districts favor the proposal, because all values in the interval are positive.

• It is likely that more voters in district H favor the proposal than in district G, because all values in the interval are positive.

• It is likely that more voters in district G favor the proposal than in district H, because all values in the interval are positive.

• It is likely that less than half the voters in both districts favor the proposal, because all values in the interval are less than 0.5.

• It is likely that there is no difference between the districts in voters who favor the proposal, because 0 is not contained in the interval.

Answer explanation

Correct. The confidence interval was created from the sample statistic of the proportion in favor of the proposal from district G minus the proportion in favor of the proposal from district H. Because all values of the difference are positive, it is likely that the population proportion from G is greater than that from H.

Machines at a bottling plant are set to fill bottles to 12 ounces. The quality control officer at the plant periodically tests the machines to be sure that the bottles are filled to an appropriate amount. The null hypothesis of the test is that the mean is at least 12 ounces. The alternative hypothesis is that the mean is less than 12 ounces.

Which of the following describes a Type I error that could result from the test?

• The test does not provide convincing evidence that the mean is less than 12 ounces, but the actual mean is at least 12 ounces.

• The test does not provide convincing evidence that the mean is less than 12 ounces, but the actual mean is less than 12 ounces

• The test does not provide convincing evidence that the mean is less than 12 ounces, but the actual mean is 12 ounces.

• The test provides convincing evidence that the mean is less than 12 ounces, but the actual mean is at least 12 ounces.

• The test provides convincing evidence that the mean is less than 12 ounces, but the actual mean is 11 ounces.

Answer explanation

Correct. A Type error occurs if the test finds evidence for an alternative hypothesis that is actually false. In this case, the actual mean fill amount is at least 12 ounces, but the test indicates the mean is less than 12 ounces.

A fair die has its faces numbered from 1 to 6. Let random variable F represent the number landing face up when the die is tossed. The probability distribution for the random variable has mean 3.5 and standard deviation 1.7078. Consider a simulation with 400 trials designed to estimate the sampling distribution of the sample mean for 5 tosses of the die. For each trial, the die is tossed 5 times, and the mean of the 5 values landing face up is recorded. The mean and standard deviation of the results of the simulation should be close to which of the following?

• Mean 3.5 and standard deviation 1.7078

• Mean 3.5 and standard deviation 0.7638

• Mean 3.5 and standard deviation 0.0854

• Mean 17.5 and standard deviation 1.7078

• Mean 17.5 and standard deviation 0.7638

Answer explanation

An experiment was conducted in which planks of wood painted red and green were shown to pigeons to investigate a pigeon’s ability to select a certain color. Pigeons could accurately select the color of the plank of wood 20 percent of the time. A simulation was conducted in which a trial consisted of a pigeon being shown eight planks of wood and its number of successes being recorded. This process was repeated many times, and the results are shown in the histogram.

Based on the results of the simulation, which of the following is closest to the probability that there were at most three successes in a trial?

• 0.06

• 0.15

• 0.21

• 0.79

• 0.94

Answer explanation

Correct. The bars corresponding to 0, 1, 2, and 3 represent at most three successes. The sum of the heights of the bars corresponds to a probability of approximately 0.94.