AP Stats Chapter 9

An environmental scientist wants to test the null hypothesis that an antipollution device for cars is not effective. Under which of the following conditions would a Type I error be committed?

A. The scientist concludes that the antipollution device is effective when it actually is not.

In a test of the null hypothesis H0: µ = 10 against the alternative hypothesis Ha: µ > 10, a sample from a normal population produces a mean of 13.4. The z-score for the sample is 2.12 and the p-value is 0.017. Based on these statistics, which of the following conclusions could be drawn?

A. There is reason to conclude that µ > 10.

The following list shows three factors that can either increase or decrease the probability of a Type IIII error. The sample size is increased. The significance level is increased. The standard error is increased. Which factors alone will cause the probability of a Type IIII error to increase?

III only

The process of producing pain-reliever tablets yields tablets with varying amounts of the active ingredient. It is claimed that the average amount of active ingredient per tablet is at least 200 milligrams. The Consumer Watchdog Bureau tests a random sample of 70 tablets. The mean content of the active ingredient for this sample is 194.3 milligrams, while the standard deviation is 21 milligrams. What is the approximate p-value for the appropriate test?

0.012

Makers of a new pain-relieving medication claim that it relieves chronic pain faster than the current top-selling pain reliever on the market. A double-blind experiment was conducted in which 10 people who experience chronic pain were randomly selected to take either the new or the current medication. Each of the 10 people recorded the time, in minutes, from taking the medication until pain relief. After an appropriate time period, each of the 10 people took the other medication and recorded the time from taking the medication until pain relief. The medication each person took first was randomly determined, and because both medications look the same, the people in the study did not know which medication was taken first. The table below shows summary statistics for the results.

0.1802

The p-value for a one-sided t-test is 0.10. If the test had been two-sided, what would the p-value have been?

0.20

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?

0.35

Suppose that on a hypothesis test for a single population mean, Ha: μ < 10. Assume that Ha is true. For a fixed sample size and significance level α, the power of the test will be greatest if the actual mean is which of the following?

8

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?

A

A survey of a random sample of 1,045 young adults found that 60 percent do not have a landline telephone number. A hypothesis test will be used to determine whether the data provide convincing statistical evidence that more than 50 percent of all young adults do not have a landline telephone number. Which of the following is the test statistic for the appropriate test?

C

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?

B

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?

B

Which of the following is the best interpretation of the power of a significance test?

B

A two-sided t-test for a population mean is conducted of the null hypothesis H0 : μ = 100. If a 90 percent t-interval constructed from the same sample data contains the value of 100, which of the following can be concluded about the test at a significance level of a = 0.10 ?

D. The p-value is greater than 0.10, and H0 should not be rejected.

A state study on labor reported that one-third of full-time teachers in the state also worked part time at another job. For those teachers, the average number of hours worked per week at the part-time job was 1313. After an increase in state teacher salaries, a random sample of 400400 teachers who worked part time at another job was selected. The average number of hours worked per week at the part-time job for the teachers in the sample was 12.512.5 with standard deviation 6.56.5 hours. Is there convincing statistical evidence, at the level of α=0.05α=0.05, that the average number of hours worked per week at part-time jobs decreased after the salary increase?

No. The p-value of the appropriate test is greater than 0.050.05.

An experimenter conducted a two-tailed hypothesis test on a set of data and obtained a p-value of 0.44. If the experimenter had conducted a one-tailed test on the same set of data, which of the following is true about the possible p-value(s) that the experimenter could have obtained?

The possible p-values are 0.22 and 0.78.

If all else is constant, which of the following results in an increase in the probability of a Type IIII error?

The sample size is decreased.

If all else is constant, which of the following would result in a decrease of the probability of a Type II error?

The sample size is increased.

A test of the hypotheses H0 : µ = 0 versus Ha : µ > 0 was conducted using a sample of size 7. The test statistic was t = 1.935. Which of the following is closest to the p-value of the test?

0.0506

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?

0.0628

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?

A

Researchers for a company that manufactures batteries want to test the hypothesis that the mean battery life of their new battery is greater than the known mean battery life of their older version. The researchers selected random samples of 32 of the new batteries, subjected the batteries to continuous use, and determined the mean and standard deviation of the battery lives in the sample. Which of the following is an appropriate test for the researchers' hypothesis?

A one-sample t-test for a population mean

Medical researchers are testing a new surgical procedure designed to minimize the side effects of surgery. The null hypothesis is that the procedure is not effective in minimizing side effects. For the researchers, the more consequential error would be that the procedure actually is effective in minimizing the side effects, but the test does not detect the effectiveness of the procedure.

A. Increase the significance level to increase the probability of a Type II error.

To obtain certification for a certain occupation, candidates take a proficiency exam. The exam consists of two sections, and neither section should be more difficult than the other. To investigate whether one section of the exam was more difficult than the other, a random sample of 50 candidates was selected. The candidates took the exam and their scores on each section were recorded. The table shows the summary statistics.

A. t=75−65/8/√50

Studies indicate that about 10 percent of polar bears weigh more than 1,000 pounds. A biologist studying the bears thinks that percent might be too high. From a random sample of polar bears, the biologist found only 8 percent of the sample weighing over 1,000 pounds. Which of the following is the most appropriate method for the biologist's study?

B. A one-sample z-test for a population proportion

A research organization reported that 4141 percent of adults who were asked to describe their day responded that they were having a good day rather than a typical day or a bad day. To investigate whether the percent would be different for high school students, 600600 high school students were randomly selected. When asked to describe their day, 245245 students reported that they were having a good day rather than a typical day or a bad day. Do the data provide convincing statistical evidence that the proportion of all high school students who would respond that they were having a good day is different from 0.410.41 ?

B. No, because the p-value is greater than any reasonable significance level.

A significance test is conducted for which the alternative hypothesis states that more than 83 percent of all adult sea turtles on a certain beach are female. A random sample of adult sea turtles from the beach found that 85 percent of the sea turtles were female. The p-value for the test is 0.4058. If the null hypothesis is true, which of the following statements is a correct interpretation of the p-value?

B. Of all possible samples of the same size, 40.58 percent will result in 85 percent or more of adult sea turtles on the beach being female.

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?

B. The population proportion of adults who watch 15 or fewer hours of television per week is 0.30.

The president of a large company recommends that employees perform, on average, 24 hours of community service each year. The president believes that the mean number of hours of community service performed last year was different from the recommended 24 hours. To estimate the mean number of hours of community service performed last year, the president obtained data from a random sample of employees and used the data to construct the 95 percent confidence interval (20.37, 23.49). If all conditions for inference were met, does the interval provide convincing statistical evidence, at a level of significance of α = 0.05, to support the president's belief that the mean number of hours of community service performed last year is different from what is recommended?

B. Yes, the interval supports the president's belief because 24 is not contained in the interval.

A machine is designed to dispense at least 12 ounces of a beverage into a bottle. To test whether the machine is working properly, a random sample of 50 bottles was selected and the mean number of ounces for the 50 bottles was computed. A test of the hypotheses H0 : µ = 12 versus Ha : µ < 12 was conducted, where µ represents the population mean number of ounces of the beverage dispensed per bottle by the machine. The p-value for the test was 0.08. Which of the following is the most appropriate conclusion to draw at the significance level of α = 0.05?

C

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=0.6H0:p=0.6 versus Ha:p>0.6Ha:p>0.6. The p-value of the test was 0.015. Which of the following is a correct interpretation of the p-value?

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.

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.

For a one-sample test for a population proportion pp and sample size n, why is it necessary that np0 and n(1−p0) are both at least 10 ?

C. The sample size must be large enough to support an assumption that the sampling distribution of the sample proportion is approximately normal.

A large clinical trial was designed to determine whether a certain vitamin improves the general health of adults. The investigators first identified 85 variables that measure various aspects of the general health of adults. Because each adult in the clinical trial was to serve as his or her own control, the 85 variables were measured for each adult, both before taking the vitamin and after taking the vitamin for three months. The investigators then performed 85 matched-pair t-tests, one for each variable. They found statistically significant results at the 0.05 level in 2 of the variables, both in the direction of improved general health. Which of the following should the investigators conclude?

C. There is insufficient evidence that the vitamin improves the health of adults because at the 0.05 significance level, one could easily get statistically significant results in 2 out of 85 tests just due to chance variability.

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=0.08H0:p=0.08 versus Ha:p>0.08Ha:p>0.08 at the significance level of α=0.01.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?

D

In a physics experiment, two different methods were used to measure the angle of deflection when a subatomic particle collides with a certain material. Ten specimens of the material were used to compare the two methods. For each specimen, the angle of deflection was measured using both methods. For each specimen, the method used first was determined by the flip of a fair coin. The difference between the measured angles was calculated for each specimen. A test of the hypothesis that the population mean difference is zero had a p-value of 0.082. The hypothesis test described had three components: the number of specimens, the sample standard deviation of the differences, and the magnitude of the sample mean difference. Compared to the test described, which of the following would have resulted in a smaller p-value?

D

A researcher's hypothesis is that the average length of salmon returning to spawn from an Alaskan river is less than the historical average length of 24 inches. The researcher collects a random sample of 45 salmon, measures the length of each fish, and computes an average length of 22 inches, with a standard deviation of 3.1 inches.

D. A one-sample t-test for a population mean

An amateur men's swimming association is trying to decide whether times in the 100-meter breaststroke will be reduced if the men shave their heads. From the population of swimmers, six were selected at random and agreed to swim two races—one before shaving their heads and one after shaving their heads. The results for each race, with times in seconds, are given in the table below.

D. A paired t-test with Ha : µd > 0

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.

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.

Educators are testing a new program designed to help children improve their reading skills. The null hypothesis of the test is that the program does not help children improve their reading skills. For the educators, the more consequential error would be that the program does not help children improve their reading skills but the test indicated that it does help.

D. Decrease the significance level to decrease the probability of Type II error.

For a certain brand of tomato seeds, the seed package claims that it takes 87 days after planting for the tomato plants to produce fruit. Sarah, a botanist, wanted to know whether the mean number of days for the plants to produce fruit where she lives is different from 87 days. She planted 40 seeds and recorded the number of days for each plant to produce fruit. With all conditions for inference met, the hypothesis test was conducted at the significance level α=0.05α=0.05, and the test resulted in a p-value of 0.0752.

D. Sarah does not have convincing statistical evidence to conclude that the population mean number of days for the plants to produce fruit is different from 87 days.

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=0.05α=0.05, and the test produced a p-value of 0.0000. Which of the following is an appropriate conclusion?

D. The researcher has statistical evidence to conclude that the population mean shower length for high school students is greater than 10 minutes.

According to a recent report, customers who shop at a certain online store spend, on average, $1,500 a year at the store. To investigate whether the mean amount spent was greater than the reported average, an economist obtained the mean and standard deviation of the amount spent in the past year by a random sample of 120 customers who shop at the store. With all conditions for inference met, the economist conducted the appropriate hypothesis test and obtained a p-value of 0.25. Which of the following statements is the most appropriate conclusion for the investigation?

D. There is not convincing statistical evidence that the mean amount of money spent each year by all customers who shop at the store is greater than $1,500.

A bank manager wants the average time that a customer waits in line to be at most 3 minutes. Customers at the bank have complained about the long wait times. To test whether the average wait time at the bank is greater than 3 minutes, 60 customers were randomly selected as they entered the bank and their wait times were recorded. The mean wait time was 4.7 minutes. A one-sample t-test resulted in a p-value of 0.00031.

E

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.

E

A group of students wanted to investigate the claim that the average number of text messages sent yesterday by students in their school was greater than 100. They asked each student in a random sample of 50 students how many text messages he or she sent yesterday. An appropriate t-test was conducted and resulted in a p-value of 0.0853. Assuming the conditions for the t-test were met, which of the following is an appropriate conclusion?

E

In 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?

E

A researcher conducted a t-test of the hypotheses H0:μ=38H0:μ=38 versus Ha:μ≠38Ha:μ≠38. The sample mean was 3535 and the p-value for the test was 0.06270.0627. What would the p-value have been if the researcher had used Ha:μ<38Ha:μ<38 as the alternative hypothesis?

E. 1/2(0.0627)

A recent study indicated that 17 percent of adults in the country actively seek out science news sites to keep current on topics in science. A university researcher believes that percent is too low. From a random sample of adults in the country, the researcher found that 22 percent of the sample actively seek out science news sites. Which of the following is the most appropriate method for the researcher's study?

E. A one-sample z-test for a population proportion

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¯Dx¯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?

E. H0:μD=0 Ha:μD<0

A representative of a car manufacturer in the United States made the following claim in a news report. Ten years ago, only 53 percent of Americans owned American-made cars, but that figure is significantly higher today. A research group conducted a study to investigate whether the claim was true. The group found that 56 percent of a randomly selected sample of car owners in the United States owned American-made cars. A test of the appropriate hypotheses resulted in a p-value of 0.283. Assuming the conditions for inference were met, is there sufficient evidence to conclude, at the significance level of a = 0.05, that the proportion of all car owners in the United States who own American-made cars has increased from what it was ten years ago?

E. No, because 0.283 > 0.05.

For a school fund-raiser, 600 raffle tickets were sold by students at the school, of which 88 were sold by one student, Audrey. Of the 600 tickets sold, 30 were randomly selected to receive prizes, and 7 of the 30 tickets selected were tickets sold by Audrey. To investigate how likely it was by chance alone that at least 7 of the 30 selected tickets could have been sold by Audrey, students in a statistics class ran a simulation. One trial of the simulation is described by the following steps. Step 1: From 600 chips, assign 88 red and the rest blue. Step 2: Select 30 chips at random without replacement. Step 3: Record the number of red chips in the selection of 30. The results of 1,000 trials of the simulation are shown in the histogram.

E. No, because the simulation suggests that Audrey selling at least 7 of 30 selected tickets would occur about 13.8% of the time.

A university will add fruit juice vending machines to its classroom buildings if the student body president is convinced that more than 20 percent of the students will use them. A random sample of n students will be selected and asked whether or not they would use the vending machines. A large-sample test for proportions at the significance level of α = 0.05 will be performed. The null hypothesis that the proportion of all students who would use the vending machines is 20 percent will be tested against the alternative that more than 20 percent of all students would use them. For which of the following situations would the power of the test be highest?

E. The sample size is n = 1,000, and 50 percent of all students use the vending machines.

A company produces millions of 1-pound packages of bacon every week. Company specifications allow for no more than 3 percent of the 1-pound packages to be underweight. To investigate compliance with the specifications, the company's quality control manager selected a random sample of 1,000 packages produced in one week and found 40 packages, or 4 percent, to be underweight. Assuming all conditions for inference are met, do the data provide convincing statistical evidence at the significance level of α = 0.05 that more than 3 percent of all the packages produced in one week are underweight?

Yes, because the p-value of 0.032 is less than the significance level of 0.05.