CHAPTER 9 REVIEW AP CLASSROOM
Consider a one-sample two-sided z-test for a population proportion. Given that conditions for inference are met, which of the following is closest to the p-value for a test statistic of z=−1.86 ?
CORRECT ANSWER: 0.0628 MY ANSWER: 0.9371
A recent study reported the mean body mass index (BMI) for adults in the United States was 26.8. A researcher believes the mean BMI of marathon runners is less than 26.8. A random sample of 35 marathon runners had a mean BMI of 22.7 with a standard deviation of 3.1. The researcher will conduct a one-sample t-test for a population mean.
CORRECT ANSWER: A Yes, all conditions have been met. MY ANSWER: C No, because marathon runners are not a representative sample of all adults in the United States.
A company that ships glass for a glass manufacturer claimed that its shipping boxes are constructed so that no more than 8 percent of the boxes arrive with broken glass. The glass manufacturer believed the actual percent is greater than 8 percent. The manufacturer selected a random sample of boxes and recorded the proportion of boxes that arrived with broken glass. The manufacturer tested the hypotheses H0:p=0.08 versus Ha:p>0.08 at the significance level of α=0.01. The test yielded a p-value of 0.001. Assuming all conditions for inference were met, which of the following is the correct conclusion?
CORRECT ANSWER: D The pp-value is less than αα, and the null hypothesis is rejected. There is convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08. MY ANSWER: B The pp-value is greater than αα, and the null hypothesis is rejected. There is not convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
To test the effectiveness of an exercise program in reducing high blood pressure, 15 participants had their blood pressures recorded before beginning the program and again after completing the program. The difference (after minus before) in blood pressure was recorded for each participant, and the sample mean difference x¯D was calculated. A hypothesis test will be conducted to investigate whether there is convincing statistical evidence for a reduction in blood pressure for all who complete the program. Which of the following is the correct set of hypotheses?
CORRECT ANSWER: E H0:μD=0Ha:μD<0MY ANSWER:B H0:x¯D=0Ha:x¯D<0
In high school X, approximately 9 percent of the students saw a certain movie on opening night. From a random sample of 200 students from high school Y, 22 saw the movie on opening night. Consider a hypothesis test to investigate whether the proportion of all students in high school Y who saw the movie on opening night is greater than that of high school X. Which of the following is the standard deviation used to calculate the test statistic for the one-sample z-test?
CORRECT ANSWER:0.09)(0.91)200−−−−−−−−√(0.09)(0.91)200 MY ANSWER: 0.09)(0.91)200−−−−−−−−√(0.09)(0.91)200
Consider a hypothesis test in which the significance level is α=0.05 and the power of the test is 0.65. What is the probability of making a Type II error?
CORRECT ANSWER:0.35 MY ANSWER: 0.95
At a large company, employees can take a course to become certified to perform certain tasks. There is an exam at the end of the course that needs to be passed for certification. The current pass rate is 0.7, but a new program is being tested to help increase the pass rate. The null hypothesis of the test is that the pass rate for the new program is 0.7. The alternative is that the pass rate for the new program is greater than 0.7. Which of the following describes a Type II error that could result from the test?
CORRECT ANSWER:A The test does not provide convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.8 MY ANSWER: A The test does not provide convincing evidence that the pass rate is greater than 0.7, but the actual pass rate is 0.8
The distribution of mass for United States pennies minted since 1982 is approximately normal with mean 2.5 grams. A random sample of 10 pennies minted since 1982 was selected. The sample had a mean mass of 2.47 grams and a standard deviation of 0.04 gram. The test statistic for the population mean has which of the following distributions?
CORRECT ANSWER:A tt-distribution with 9 degrees of freedom MY ANSWER:A tt-distribution with 9 degrees of freedom
A marketing agency selected a random sample of television viewers to test the claim that the proportion of viewers who watch a particular show is less than 0.20 at a level of significance of 0.05. The test yielded a p-value of 0.47. Assuming all conditions for inference were met, which of the following is the correct conclusion?
CORRECT ANSWER:At the level of significance of 0.05, the null hypothesis is not rejected. There is not convincing evidence to suggest the true proportion of television viewers who watch the show is less than 0.20. MY ANSWER: At the level of significance of 0.05, the null hypothesis is not rejected. There is not convincing evidence to suggest the true proportion of television viewers who watch the show is less than 0.20.
Consider the results of a hypothesis test, which indicate there is not enough evidence to reject the null hypothesis. Which of the following statements about error is correct?
CORRECT ANSWER:B A Type IIII error could have been made, but not a Type II error. MY ANSWER: B A Type IIII error could have been made, but not a Type II error.
Last year the mean cost μ for a one-bedroom rental in a certain city was $1,200 per month. Eli is looking for a one-bedroom apartment and is investigating whether the mean cost is less now than what it was last year. A random sample of apartments had a sample mean x¯ of $1,180 per month. Assuming all conditions for inference are met, Eli will conduct a hypothesis test as part of his investigation. Which of the following is the correct set of hypotheses?
CORRECT ANSWER:B H0:μ=1,200Ha:μ<1,200 MY ANSWER: B H0:μ=1,200Ha:μ<1,200
Which of the following is the best interpretation of the power of a significance test
CORRECT ANSWER:B Power is the probability of detecting an effect if an effect exists. MY ANSWER: B Power is the probability of detecting an effect if an effect exists.
A social scientist believed that less than 30 percent of adults in the United States watch 15 or fewer hours of television per week. To test the belief, the scientist randomly selected 1,250 adults in the United States. The sample proportion of adults who watch 15 or fewer hours of television per week was 0.28, and the resulting hypothesis test had a p-value of 0.061. The computation of the p-value assumes which of the following is true?
CORRECT ANSWER:B The population proportion of adults who watch 15 or fewer hours of television per week is 0.30. MY ANSWER: C The population proportion of adults who watch 15 or fewer hours of television per week is less than 0.30.
The germination rate is the rate at which plants begin to grow after the seed is planted. A seed company claims that the germination rate for their seeds is 90 percent. Concerned that the germination rate is actually less than 90 percent, a botanist obtained a random sample of seeds, of which only 80 percent germinated. What are the correct hypotheses for a one-sample z-test for a population proportion p ?
CORRECT ANSWER:C H0:p=0.90Ha:p<0.90H0:p=0.90Ha:p<0.90 MY ANSWER: B H0:p=0.80Ha:p>0.80
Past studies indicate that about 60 percent of the trees in a forested region are classified as softwood. A botanist studying the region suspects that the proportion might be greater than 0.60. The botanist obtained a random sample of trees from the region and conducted a test of H0:p=0.6 versus Ha:p>0.6. The p-value of the test was 0.015. Which of the following is a correct interpretation of the p-value?
CORRECT ANSWER:C If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is the probability of obtaining a sample proportion as large as or larger than the one obtained by the botanist MY ANSWER: C If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is the probability of obtaining a sample proportion as large as or larger than the one obtained by the botanist
A software company provides specialized resort reservation software that can be tailored to the needs of its customers. The company's 120 customers pay yearly subscription costs that can vary from customer to customer. The company knows that to be profitable, it needs each customer to be spending at least $23,000 per year, on average. The company selects a random sample of 33 customers and computes a mean of $27,871 and a standard deviation of $309.10. It performs a hypothesis test and computes a very small p-value. The software company concludes that the mean is greater than $23,000. Was it appropriate for the software company to perform the hypothesis test and make the conclusion that was made?
CORRECT ANSWER:C No, because the sample is more than 10 percent of the population, so one of the conditions for conducting a hypothesis test has not been met. MY ANSWER: E No, because the distribution of the sample data is skewed.
A six-week fitness program was designed to decrease the time it takes retired individuals to walk one mile. At the beginning of the program, 20 randomly selected retired individuals were invited to participate, and their times to walk a mile were recorded. After the six-week program, their times to walk a mile were again recorded. Most participants saw little to no improvement in their times to walk one mile; however, a few participants saw drastic improvements in their times to walk one mile. The program director would like to perform a hypothesis test to determine if the program reduces the mean time for retired individuals to walk a mile. Which of the following statements is true?
CORRECT ANSWER:D Because the sample size of 20 is less than 30 and the improvements in walk times in the sample data appear to be skewed, the distribution of sample means should not be assumed to be approximately normal.MY ANSWER: The sampling distribution of sample means can be assumed to be approximately normal because the distribution of the sample data is not skewed.
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?
CORRECT ANSWER:D The test provides convincing evidence that the mean is less than 12 ounces, but the actual mean is at least 12 ounces. MY ANSWER: D The test provides convincing evidence that the mean is less than 12 ounces, but the actual mean is at least 12 ounces.
Consider a population with population proportion p, and a sample from the population with sample proportion pˆ. Which of the following describes the purpose of the one-sample z-test?
CORRECT ANSWER:D To estimate the probability of observing a value as extreme as pˆp^ given p MY ANSWER: To estimate the value of pp
A one-sample z-test for a population proportion will be conducted using a simple random sample selected without replacement from a population. Which of the following is a check for independence
CORRECT ANSWER:The population size is more than 10 times the sample size. MY ANSWER: A np0≥10np0≥10 and n(1−p0)≥10n(1−p0)≥10 for sample size nn and population proportion p0p0.
A car company claims that its new car, the GoFast2000, has a gas mileage of 35 miles per gallon (mpg). A consumer group suspects that the true mean gas mileage of the new cars is less than 35 mpg. The group tests 50 randomly selected GoFast2000 cars and finds a sample mean of 34.8 mpg. With all assumptions for inference met, a hypothesis test resulted in a p-value of 0.324. For a significance level of α=0.05, which of the following is a correct conclusion?
CORRECT ANSWER:The pp-value is greater than 0.05, and the null hypothesis is not rejected. There is not convincing statistical evidence that the mean is less than 35 mpgmpg. MY ANSWER: The pp-value is less than 0.05, and the null hypothesis is not rejected. There is not convincing statistical evidence that the mean is less than 35 mpgmpg.
Most dermatologists recommend that the ideal shower lasts approximately 10 minutes. A researcher suspects that the average shower length of high school students is greater than 10 minutes. To test the belief, the researcher surveyed 125 randomly selected high school students and found that their average shower length was 14.7 minutes. With all conditions for inference met, a hypothesis test was conducted at the significance level of α=0.05, and the test produced a p-value of 0.0000. Which of the following is an appropriate conclusion?
CORRECT ANSWER:The researcher has statistical evidence to conclude that the population mean shower length for high school students is greater than 10 minutesMY ANSWER: The researcher does not have statistical evidence to conclude that the sample mean shower length for high school students is greater than 10 minutes.
If all else is constant, which of the following would result in a decrease of the probability of a Type II error?
CORRECT ANSWER:The sample size is increased. MY ANSWER: The significance level is decreased.
Which of the following gives the probability of making a Type I error?
CORRECT ANSWER:The significance level MY ANSWER: The significance level
For a certain brand of canned corn, the company claims that the mean weight of the contents of the cans is 15.25 ounces. A random sample of 36 cans were selected. The sample was found to have mean 15.18 ounces and standard deviation 0.12 ounce. A hypothesis test will be conducted to investigate whether there is evidence to support the belief that the mean is less than 15.25 ounces. Which of the following is the correct test statistic for the hypothesis test?
CORRECT ANSWER:t=15.18−15.250.126MY ANSWER: B t=15.18−15.250.1236
n a population of bats living in a certain region, 30 percent have a wingspan greater than 10 inches. In a random sample of 80 bats living outside of the region, 20 had a wingspan greater than 10 inches. Consider a one-sample z-test to investigate whether there is evidence that the proportion of bats with a wingspan greater than 10 inches living outside the region is different from that of the bats living in the region. Which of the following is the correct test statistic?
CORRECT ANSWER:z=0.25−0.30(0.30)(0.70)80√ MY ANSWER: z=0.25−0.30(0.30)(0.70)80√