3. P-value. A medical researcher tested a new treatment

1. One sided or two? In each of the following situations, is the alternative hypothesis one-sided or two-sided? What are the hypotheses? a) A business student conducts a taste test to see whether students prefer Diet Coke or Diet Pepsi. b) PepsiCo recently reformulated Diet Pepsi in an attempt to appeal to teenagers. They run a taste test to see if the new formula appeals to more teenagers than the standard formula. c) A budget override in a small town requires a twothirds majority to pass. A local newspaper conducts a poll to see if theres evidence it will pass. d) One financial theory states that the stock market will go up or down with equal probability. A student collects data over several years to test the theory.

2. Which alternative? In each of the following situations, is the alternative hypothesis one-sided or two-sided? What are the hypotheses? a) A college dining service conducts a survey to see if students prefer plastic or metal cutlery. b) In recent years, 10% of college juniors have applied for study abroad. The deans office conducts a survey to see if thats changed this year. c) A pharmaceutical company conducts a clinical trial to see if more patients who take a new drug experience headache relief than the 22% who claimed relief after taking the placebo. d) At a small computer peripherals company, only 60% of the hard drives produced passed all their performance tests the first time. Management recently invested a lot of resources into the production system and now conducts a test to see if it helped.

3. P-value. A medical researcher tested a new treatment for poison ivy against the traditional ointment. He concluded that the new treatment is more effective. Explain what the P-value of 0.047 means in this context.

4. Another P-value. Have harsher penalties and ad campaigns increased seat-belt use among drivers and passengers? Observations of commuter traffic failed to find evidence of a significant change compared with three years ago. Explain what the study's P-value of 0.17 means in this context.

5. Alpha. A researcher developing scanners to search for hidden weapons at airports has concluded that a new device is significantly better than the current scanner. He made this decision based on a test using a = 0.05. Would he have made the same decision at a = 0.10? How about a = 0.01? Explain.

6. Alpha again. Environmentalists concerned about the impact of high-frequency radio transmissions on birds found that there was no evidence of a higher mortality rate among hatchlings in nests near cell towers. They based this conclusion on a test using a = 0.05. Would they have made the same decision at a = 0.10? How about a = 0.01? Explain.

7. Significant? Public health officials believe that 90% of children have been vaccinated against measles. A random survey of medical records at many schools across the country found that, among more than 13,000 children, only 89.4% had been vaccinated. A statistician would reject the 90% hypothesis with a P-value of P = 0.011. a) Explain what the P-value means in this context. b) The result is statistically significant, but is it important? Comment.

8. Significant again? A new reading program may reduce the number of elementary school students who read below grade level. The company that developed this program supplied materials and teacher training for a large-scale test involving nearly 8500 children in several different school districts. Statistical analysis of the results showed that the percentage of students who did not meet the grade-level goal was reduced from 15.9% to 15.1%. The hypothesis that the new reading program produced no improvement was rejected with a P-value of 0.023. a) Explain what the P-value means in this context. b) Even though this reading method has been shown to be significantly better, why might you not recommend that your local school adopt it?

9. Success. In August 2004, Time magazine reported the results of a random telephone poll commissioned by the Spike network. Of the 1302 men who responded, only 39 said that their most important measure of success was their work. a) Estimate the percentage of all American males who measure success primarily from their work. Use a 98% confidence interval. Check the conditions first. b) Some believe that few contemporary men judge their success primarily by their work. Suppose we wished to conduct a hypothesis test to see if the fraction has fallen below the 5% mark. What does your confidence interval indicate? Explain. c) What is the level of significance of this test? Explain.

10. Is the Euro fair? Soon after the Euro was introduced as currency in Europe, it was widely reported that someone had spun a Euro coin 250 times and gotten heads 140 times. We wish to test a hypothesis about the fairness of spinning the coin. a) Estimate the true proportion of heads. Use a 95% confidence interval. Dont forget to check the conditions. b) Does your confidence interval provide evidence that the coin is unfair when spun? Explain. c) What is the significance level of this test? Explain.

11. Approval 2007. In May 2007, George W. Bushs approval rating stood at 30% according to a CBS News/New York Times national survey of 1125 randomly selected adults. a) Make a 95% confidence interval for his approval rating by all U.S. adults. b) Based on the confidence interval, test the null hypothesis that Bushs approval rating was no better than the 27% level established by Richard Nixon during the Watergate scandal.

12. Superdads. The Spike network commissioned a telephone poll of randomly sampled U.S. men. Of the 712 respondents who had children, 22% said yes to the question Are you a stay-at-home dad? (Time, August 23, 2004) a) To help market commercial time, Spike wants an accurate estimate of the true percentage of stay-at-home dads. Construct a 95% confidence interval. b) An advertiser of baby-carrying slings for dads will buy commercial time if at least 25% of men are stayat- home dads. Use your confidence interval to test an appropriate hypothesis, and make a recommendation to the advertiser. c) Could Spike claim to the advertiser that it is possible that 25% of men with young children are stay-at-home dads? What is wrong with the reasoning?

13. Dogs. Canine hip dysplasia is a degenerative disease that causes pain in many dogs. Sometimes advanced warning signs appear in puppies as young as 6 months. A veterinarian checked 42 puppies whose owners brought them to a vaccination clinic, and she found 5 with early hip dysplasia. She considers this group to be a random sample of all puppies. a) Explain we cannot use this information to construct a confidence interval for the rate of occurrence of early hip dysplasia among all 6-month-old puppies. *b) Construct a plus-four confidence interval and interpret it in this context.

14. Fans. A survey of 81 randomly selected people standing in line to enter a football game found that 73 of them were home team fans. a) Explain why we cannot use this information to construct a confidence interval for the proportion of all people at the game who are fans of the home team. *b) Construct a plus-four confidence interval and interpret it in this context.

15. Loans. Before lending someone money, banks must decide whether they believe the applicant will repay the loan. One strategy used is a point system. Loan officers assess information about the applicant, totaling points they award for the persons income level, credit history, current debt burden, and so on. The higher the point total, the more convinced the bank is that its safe to make the loan. Any applicant with a lower point total than a certain cutoff score is denied a loan. We can think of this decision as a hypothesis test. Since the bank makes its profit from the interest collected on repaid loans, their null hypothesis is that the applicant will repay the loan and therefore should get the money. Only if the persons score falls below the minimum cutoff will the bank reject the null and deny the loan. This system is reasonably reliable, but, of course, sometimes there are mistakes. a) When a person defaults on a loan, which type of error did the bank make? b) Which kind of error is it when the bank misses an opportunity to make a loan to someone who would have repaid it? c) Suppose the bank decides to lower the cutoff score from 250 points to 200. Is that analogous to choosing a higher or lower value of for a hypothesis test? Explain. d) What impact does this change in the cutoff value have on the chance of each type of error?

16. Spam. Spam filters try to sort your e-mails, deciding which are real messages and which are unwanted. One method used is a point system. The filter reads each incoming e-mail and assigns points to the sender, the subject, key words in the message, and so on. The higher the point total, the more likely it is that the message is unwanted. The filter has a cutoff value for the point total; any message rated lower than that cutoff passes through to your inbox, and the rest, suspected to be spam, are diverted to the junk mailbox. We can think of the filters decision as a hypothesis test. The null hypothesis is that the e-mail is a real message and should go to your inbox. A higher point total provides evidence that the message may be spam; when theres sufficient evidence, the filter rejects the null, classifying the message as junk. This usually works pretty well, but, of course, sometimes the filter makes a mistake. a) When the filter allows spam to slip through into your inbox, which kind of error is that? b) Which kind of error is it when a real message gets classified as junk? c) Some filters allow the user (thats you) to adjust the cutoff. Suppose your filter has a default cutoff of 50 points, but you reset it to 60. Is that analogous to choosing a higher or lower value of for a hypothesis test? Explain. d) What impact does this change in the cutoff value have on the chance of each type of error?

17. Second loan. Exercise 15 describes the loan score method a bank uses to decide which applicants it will lend money. Only if the total points awarded for various aspects of an applicants financial condition fail to add up to a minimum cutoff score set by the bank will the loan be denied. a) In this context, what is meant by the power of the test? b) What could the bank do to increase the power? c) Whats the disadvantage of doing that?

18. More spam. Consider again the points-based spam filter described in Exercise 16. When the points assigned to various components of an e-mail exceed the cutoff value youve set, the filter rejects its null hypothesis (that the message is real) and diverts that e-mail to a junk mailbox. a) In this context, what is meant by the power of the test? b) What could you do to increase the filters power? c) Whats the disadvantage of doing that?

19. Homeowners 2005. In 2005 the U.S. Census Bureau reported that 68.9% of American families owned their homes. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers in order to encourage people to become homeowners. They decide to adopt the plan on a 2-year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. a) In words, what will their hypotheses be? b) What would a Type I error be? c) What would a Type II error be? d) For each type of error, tell who would be harmed. e) What would the power of the test represent in this context?

20. Alzheimers. Testing for Alzheimers disease can be a long and expensive process, consisting of lengthy tests and medical diagnosis. Recently, a group of researchers (Solomon et al., 1998) devised a 7-minute test to serve as a quick screen for the disease for use in the general population of senior citizens. A patient who tested positive would then go through the more expensive battery of tests and medical diagnosis. The authors reported a false positive rate of 4% and a false negative rate of 8%. a) Put this in the context of a hypothesis test. What are the null and alternative hypotheses? b) What would a Type I error mean? c) What would a Type II error mean? d) Which is worse here, a Type I or Type II error? Explain. e) What is the power of this test?

21. Testing cars. Aclean air standard requires that vehicle exhaust emissions not exceed specified limits for various pollutants. Many states require that cars be tested annually to be sure they meet these standards. Suppose state regulators double-check a random sample of cars that a suspect repair shop has certified as okay. They will revoke the shops license if they find significant evidence that the shop is certifying vehicles that do not meet standards. a) In this context, what is a Type I error? b) In this context, what is a Type II error? c) Which type of error would the shops owner consider more serious? d) Which type of error might environmentalists consider more serious?

22. Quality control. Production managers on an assembly line must monitor the output to be sure that the level of defective products remains small. They periodically inspect a random sample of the items produced. If they find a significant increase in the proportion of items that must be rejected, they will halt the assembly process until the problem can be identified and repaired. a) In this context, what is a Type I error? b) In this context, what is a Type II error? c) Which type of error would the factory owner consider more serious? d) Which type of error might customers consider more serious?

23. Cars again. As in Exercise 21, state regulators are checking up on repair shops to see if they are certifying vehicles that do not meet pollution standards. a) In this context, what is meant by the power of the test the regulators are conducting? b) Will the power be greater if they test 20 or 40 cars? Why? c) Will the power be greater if they use a 5% or a 10% level of significance? Why? d) Will the power be greater if the repair shops inspectors are only a little out of compliance or a lot? Why?

24. Production. Consider again the task of the quality control inspectors in Exercise 22. a) In this context, what is meant by the power of the test the inspectors conduct? b) They are currently testing 5 items each hour. Someone has proposed that they test 10 instead. What are the advantages and disadvantages of such a change? c) Their test currently uses a 5% level of significance. What are the advantages and disadvantages of changing to an alpha level of 1%? d) Suppose that, as a day passes, one of the machines on the assembly line produces more and more items that are defective. How will this affect the power of the test?

25. Equal opportunity? A company is sued for job discrimination because only 19% of the newly hired candidates were minorities when 27% of all applicants were minorities. Is this strong evidence that the companys hiring practices are discriminatory? a) Is this a one-tailed or a two-tailed test? Why? b) In this context, what would a Type I error be? c) In this context, what would a Type II error be? d) In this context, what is meant by the power of the test? e) If the hypothesis is tested at the 5% level of significance instead of 1%, how will this affect the power of the test? f) The lawsuit is based on the hiring of 37 employees. Is the power of the test higher than, lower than, or the same as it would be if it were based on 87 hires?

26. Stop signs. Highway safety engineers test new road signs, hoping that increased reflectivity will make them more visible to drivers. Volunteers drive through a test course with several of the new- and old-style signs and rate which kind shows up the best. a) Is this a one-tailed or a two-tailed test? Why? b) In this context, what would a Type I error be? c) In this context, what would a Type II error be? d) In this context, what is meant by the power of the test? e) If the hypothesis is tested at the 1% level of significance instead of 5%, how will this affect the power of the test? f) The engineers hoped to base their decision on the reactions of 50 drivers, but time and budget constraints may force them to cut back to 20. How would this affect the power of the test? Explain.

27. Dropouts. A Statistics professor has observed that for several years about 13% of the students who initially enroll in his Introductory Statistics course withdraw before the end of the semester. A salesman suggests that he try a statistics software package that gets students more involved with computers, predicting that it will cut the dropout rate. The software is expensive, and the salesman offers to let the professor use it for a semester to see if the dropout rate goes down significantly. The professor will have to pay for the software only if he chooses to continue using it. a) Is this a one-tailed or two-tailed test? Explain. b) Write the null and alternative hypotheses. c) In this context, explain what would happen if the professor makes a Type I error. d) In this context, explain what would happen if the professor makes a Type II error. e) What is meant by the power of this test?

28. Ads. Acompany is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad and recognize the companys product. The radio station conducts a random phone survey of 400 people. a) What are the hypotheses? b) The station plans to conduct this test using a 10% level of significance, but the company wants the significance level lowered to 5%. Why? c) What is meant by the power of this test? d) For which level of significance will the power of this test be higher? Why? e) They finally agree to use , but the company proposes that the station call 600 people instead of the 400 initially proposed. Will that make the risk of Type II error higher or lower? Explain.

29. Dropouts, part II. Initially, 203 students signed up for the Stats course in Exercise 27. They used the software suggested by the salesman, and only 11 dropped out of the course. a) Should the professor spend the money for this software? Support your recommendation with an appropriate test. b) Explain what your P-value means in this context.

30. Testing the ads. The company in Exercise 28 contacts 600 people selected at random, and only 133 remember the ad. a) Should the company renew the contract? Support your recommendation with an appropriate test. b) Explain what your P-value means in this context.

31. Two coins. In a drawer are two coins. They look the same, but one coin produces heads 90% of the time when spun while the other one produces heads only 30% of the time. You select one of the coins. You are allowed to spin it once and then must decide whether the coin is the 90%- or the 30%-head coin. Your null hypothesis is that your coin produces 90% heads. a) What is the alternative hypothesis? b) Given that the outcome of your spin is tails, what would you decide? What if it were heads? c) How large is in this case? d) How large is the power of this test? (Hint: How many possibilities are in the alternative hypothesis?) e) How could you lower the probability of a Type I error and increase the power of the test at the same time?

32. Faulty or not? You are in charge of shipping computers to customers. You learn that a faulty disk drive was put into some of the machines. Theres a simple test you can perform, but its not perfect. All but 4% of the time, a good disk drive passes the test, but unfortunately, 35% of the bad disk drives pass the test, too. You have to decide on the basis of one test whether the disk drive is good or bad. Make this a hypothesis test. a) What are the null and alternative hypotheses? b) Given that a computer fails the test, what would you decide? What if it passes the test? c) How large is for this test? d) What is the power of this test? (Hint: How many possibilities are in the alternative hypothesis?)

33. Hoops. A basketball player with a poor foul-shot record practices intensively during the off-season. He tells the coach that he has raised his proficiency from 60% to 80%. Dubious, the coach asks him to take 10 shots, and is surprised when the player hits 9 out of 10. Did the player prove that he has improved? a) Suppose the player really is no better than before still a 60% shooter. Whats the probability he can hit at least 9 of 10 shots anyway? (Hint: Use a Binomial model.) b) If that is what happened, now the coach thinks the player has improved when he has not. Which type of error is that? c) If the player really can hit 80% now, and it takes at least 9 out of 10 successful shots to convince the coach, whats the power of the test? d) List two ways the coach and player could increase the power to detect any improvement.

34. Pottery. An artist experimenting with clay to create pottery with a special texture has been experiencing difficulty with these special pieces. About 40% break in the kiln during firing. Hoping to solve this problem, she buys some more expensive clay from another supplier. She plans to make and fire 10 pieces and will decide to use the new clay if at most one of them breaks. a) Suppose the new, expensive clay really is no better than her usual clay. Whats the probability that this test convinces her to use it anyway? (Hint: Use a Binomial model.) b) If she decides to switch to the new clay and it is no better, what kind of error did she commit? c) If the new clay really can reduce breakage to only 20%, whats the probability that her test will not detect the improvement? d) How can she improve the power of her test? Offer at least two suggestions.