Ap stats unit 6

The manager of a city recreation center wants to estimate the percent of city residents who favor a proposal to build a new dog park. To gather data, the manager will select a random sample of city residents. Which of the following is the most appropriate interval for the manager to use for such an estimate?

A one-sample z-interval for a population proportion

A random sample of 100 people from Country S had 15 people with blue eyes. A separate random sample of 100 people from Country B had 25 people with blue eyes. Assuming all conditions are met, which of the following is a 95 percent confidence interval to estimate the difference in population proportions of people with blue eyes (Country S minus Country B) ?

(−0.21,0.01)

A hypothesis test was conducted to investigate whether the population proportion of students at a certain college who went to the movie theater last weekend is greater than 0.2. A random sample of 100 students at this college resulted in a test statistic of 2.25. Assuming all conditions for inference were met, which of the following is closest to the p-value of the test?

0.0122

Consider a hypothesis test in which the significance level is α=0.05 and the probability of a Type II error is 0.18. What is the power of the test?

0.82

Suppose a 90 percent confidence interval to estimate a population proportion was calculated from a sample proportion of 18 percent and a margin of error of 4 percent. What is the width of the confidence interval?

8 percent

A study reported that 28 percent of middle school students in a certain state participate in community service activities. A teacher believes that the rate is greater than 28 percent for the middle school students in the teacher's district. The teacher selected a random sample of middle school students from the district, and the percent of students in the sample who participated in community service activities was found to be 32 percent. Which of the following is the most appropriate method for investigating the teacher's belief?

A one-sample z-test for a population proportion

A study reports that 75 percent of young adults in a county get their news from online sources. A sociologist believes that the percentage is actually greater than 75 percent. The sociologist will select a random sample of young adults from around the county to interview. Which of the following is the most appropriate method for investigating the sociologist's belief?

A one-sample z-test for a population proportion

At a research facility that designs rocket engines, researchers know that some engines fail to ignite as a result of fuel system error. From a random sample of 40 engines of one design, 14 failed to ignite as a result of fuel system error. From a random sample of 30 engines of a second design, 9 failed to ignite as a result of fuel system error. The researchers want to estimate the difference in the proportion of engine failures for the two designs. Which of the following is the most appropriate method to create the estimate?

A two-sample z -interval for a difference in population proportions

A yearbook company was investigating whether there is a significant difference between two states in the percents of high school students who order yearbooks. From a random sample of 150 students selected from one state, 70 had ordered a yearbook. From a random sample of 100 students selected from the other state, 65 had ordered a yearbook. Which of the following is the most appropriate method for analyzing the results?

A two-sample z -test for a difference in population proportions

A behavioral scientist investigated whether there is a significant difference in the percentages of men and women who purchase silver-colored cars. The scientist selected a random sample of 50 men and a random sample of 52 women who had recently purchased a new car. Of the men selected, 16 had purchased a silver-colored car. Of the women selected, 9 had purchased a silver-colored car. Which of the following is the most appropriate method for analyzing the results?

A two-sample z -test for the difference in population proportions

Researchers are testing a new diagnostic tool designed to identify a certain condition. The null hypothesis of the significance test is that the diagnostic tool is not effective in detecting the condition. For the researchers, the more consequential error would be that the diagnostic tool is not effective, but the significance test indicated that it is effective. Which of the following should the researchers do to avoid the more consequential error?

Decrease the significance level to decrease the probability of Type I error.

Consider a 90 percent confidence interval constructed to estimate the difference between two population proportions. Which of the following is the best interpretation of what is meant by 90 percent confidence?

For repeated random sampling from the populations with samples of the same size, approximately 90% of the confidence intervals constructed will capture the true difference between the population proportions.

Which of the following indicates that the use of a two-sample z-interval for a difference in population proportions is appropriate? I. Two populations of interest exist. II. The variable of interest is categorical. III. The intent is to estimate a difference in sample proportions.

I and II only

Molly works for a meat producer, and she needs to determine whether containers of ground beef have the correct fat content. She obtains a random sample of 120 containers of ground beef and finds that 84 percent have the correct fat content. Molly then conducts a hypothesis test of H0:p=0.80 versus Ha:p≠0.80 and calculates a test statistic of 1.10 with a p-value of 0.273. Which of the following best represents the meaning of the p-value?

If the population proportion is 0.80, the probability of observing a sample proportion of at least 0.84 or at most 0.76 is 0.273.

Independent random samples of students were taken from two high schools, R and S, and the proportion of students who drive to school in each sample was recorded. The difference between the two sample proportions (R minus S) was 0.07. Under the assumption that all conditions for inference were met, a hypothesis test was conducted where the alternative hypothesis was the population proportion of students who drive to school for R was greater than that for S. The p-value of the test was 0.114. Which of the following is the correct interpretation of the p-value?

If the proportions of all students who drive to school are the same for both high schools, the probability of observing a sample difference of at least 0.07 is 0.114.

A research group studying cell phone habits asked the question "Do you ever use your cell phone to make a payment at a convenience store?" to people selected from two random samples of cell phone users. One sample consisted of older adults, ages 35 years and older, and the other sample consisted of younger adults, ages 18 years to 34 years. The proportion of people who answered yes in each sample was used to create a 95 percent confidence interval of (0.097,0.125) to estimate the difference (younger minus older) between the population proportions of people who would answer yes to the question. Which of the following is the best description of what is meant by 95 percent confidence?

In repeated random sampling with the same sample size, approximately 95% of the intervals constructed from the samples will capture the difference in population proportions of people who would answer yes to the question.

A new drug to treat a certain condition is being tested. The null hypothesis of the test is that the drug is not effective. For the researchers, the more consequential error would be for the drug to be effective, but the test does not detect the effect. Which of the following should the researchers do to avoid the more consequential error?

Increase the significance level to increase the probability of Type I error.

A recent national survey indicated that 73 percent of respondents try to include locally grown foods in their diets. A 95 percent confidence interval for the proportion of all people in the country who try to include locally grown foods in their diets is given as (0.70,0.76). Assume all conditions for inference were met. Based on the confidence interval, which of the following claims is supported?

Less than 80 percent of all people in the country try to include locally grown foods in their diets.

A recent survey of cell phone users indicated that 56 percent of the respondents prefer to use cell phones for texting rather than for making phone calls. A 95 percent confidence interval for the estimate of all cell phone users who prefer to use cell phones for texting has a margin of error of 3 percent. Assume all conditions for inference have been met. Based on the confidence interval, which of the following claims is supported?

More than half of all people prefer texting.

A major credit card company is interested in the proportion of individuals who use a competitor's credit card. Their null hypothesis is H0: p=0.65, and based on a sample they find a sample proportion of 0.70 and a p-value of 0.053. Is there convincing statistical evidence at the 0.05 level of significance that the true proportion of individuals who use the competitor's card is actually greater than 0.65 ?

No, because the p -value 0.053 is greater than the significance level 0.05

A town council wants to estimate the proportion of residents who are in favor of a proposal to upgrade the computers in the town library. A random sample of 100 residents was selected, and 97 of those selected indicated that they were in favor of the proposal. Is it appropriate to assume that the sampling distribution of the sample proportion is approximately normal?

No, because the sample is not large enough to satisfy the normality conditions.

A marketing representative wants to estimate the proportion of people in a state who like the new design on the packaging of a certain cleaning product. The representative interviewed 100 people at a certain supermarket, and 82 people indicated that they liked the new design. Have the conditions for creating a confidence interval for the population proportion been met?

No, because the sample may not be representative of all people in the state

Is the significance level of a hypothesis test equivalent to the probability that the null hypothesis is true?

No, the significance level is the probability of rejecting the null hypothesis when the null hypothesis is actually true.

Surveys were sent to a random sample of owners of all-wheel-drive (AWD) vehicles and to a random sample of owners of front-wheel-drive (FWD) vehicles. The proportion of owners who were satisfied with their vehicles was recorded for each sample. The sample proportions were used to construct the 95 percent confidence interval for a difference in population proportions (FWD minus AWD) for satisfied owners. The interval is given as (−0.01,0.12). A car company believes that the proportion of satisfied owners of AWD vehicles differs from the proportion of satisfied owners of FWD vehicles. Does the confidence interval provide evidence that this belief is plausible?

No. The interval contains 0.

Chicken hatcheries employ workers to determine the sex of the baby chicks. The hatcheries claim that the workers are correct 95 percent of the time. An investigator believes the workers' success rate (workers are correct) is actually less than 95 percent of the time. The investigator selects a random sample of chicks and finds that the hatchery workers had a success rate of 0.936. The conditions for inference were checked and verified, and the p-value of the test was given as 0.0322. If the null hypothesis is true, which of the following statements is a correct interpretation of the p-value?

Of all possible samples of the same size, 3.22% will result in a success rate of 93.6% or less.

A newspaper article claims that 92 percent of teens use social media. To investigate the claim, a polling organization selected a random sample of 100 teens, and 96 teens in the sample indicated that they use social media. Given the data, why is it not appropriate to use a one-sample z-test for a proportion to test the newspaper's claim?

The expected number of teens in the sample who do not use social media is less than 10.

A recent increase in sales of microchips has forced a computer company to buy a new processing machine to help keep up with demand. The builders of the new machine claim that it produces fewer defective microchips than the older machine. From a random sample of 90 microchips produced on the old machine, 5 were found to be defective. From a random sample of 83 microchips produced on the new machine, 3 were found to be defective. The quality control manager wants to construct a confidence interval to estimate the difference between the proportion of defective microchips from the older machine and the proportion of defective microchips from the new machine. Why is it not appropriate to calculate a two-sample z-interval for a difference in proportions?

The normality of the sampling distribution of the difference in sample proportions cannot be established.

In a hypothesis test for a single proportion, which of the following is assumed for the calculation of the p-value?

The null hypothesis is true.

Which of the following is defined by the significance level of a hypothesis test?

The probability of Type I error

Suppose a researcher wants to use a confidence interval to estimate an unknown population proportion p. Which of the following is not a correct statement?

The probability that p is in the interval is equal to the level of confidence for the interval.

If all other factors are held constant, which of the following results in an increase in the probability of a Type II error?

The significance level is decreased.

If all other factors are held constant, which of the following results in a decrease in the probability of a Type II error?

The standard error is decreased.

Machines at a factory produce circular washers with a specified diameter. The quality control manager at the factory periodically tests a random sample of washers to be sure that greater than 90 percent of the washers are produced with the specified diameter. The null hypothesis of the test is that the proportion of all washers produced with the specified diameter is equal to 90 percent. The alternative hypothesis is that the proportion of all washers produced with the specified diameter is greater than 90 percent. Which of the following describes a Type I error that could result from the test?

The test provides convincing evidence that the proportion is greater than 90%, but the actual proportion is equal to 90%.

Sue and Javier are working on a statistics project to estimate the proportion of students at their school who have a pet dog. Sue selects a random sample of 81 students from the 2,400 students at their school, and Javier selects a separate random sample of 64 students. They will both construct a 90 percent confidence interval from their estimates. Consider the situation in which the sample proportion from Sue's sample is equal to the sample proportion from Javier's sample. Which of the following statements correctly describes their intervals?

The width of Sue's interval will be narrower than the width of Javier's interval.

A random sample of 83 residents of a certain town were asked whether they approve of a proposal to improve the town's aging bridges. The 95 percent confidence interval to estimate the proportion of all residents of the town who approve of the proposal was calculated to be (0.361,0.579). Which of the following is a correct interpretation of the interval?

We are 95 percent confident that the proportion of all residents in the town who favor the proposal is between 0.361 and 0.579.

A large company offered gym memberships to its employees as part of a program to keep employees healthy. A random sample of employees with a gym membership and a random sample of employees without a gym membership were taken, and the proportion of employees who had taken at least one sick day in the past month was recorded for each sample. A 90 percent confidence interval for the difference in population proportions (membership minus no membership) was found to be (−0.13,0.05). Employees believe that there is no difference in absenteeism between those with a gym membership and those without a gym membership. Does the confidence interval provide evidence that this belief is plausible?

Yes. The value of 0 is contained in the interval, which indicates that no difference is plausible.