Errors For each of the following situations, state whether a Type I, a Type II, or

Parameters and hypotheses For each of the following situations, define the parameter (proportion or mean) and write the null and alternative hypotheses in terms of parameter values. Example: We want to know if the proportion of up days in the stock market is 50%. Answer: Let p = the proportion of up days. Ho: p = 0.5 vs. HA: p _ 0.5. a) A casino wants to know if their slot machine really delivers the 1 in 100 win rate that it claims. b) A pharmaceutical company wonders if their new drug has a cure rate different from the 30% reported by the placebo. c) A bank wants to know if the percentage of customers using their website has changed from the 40% that used it before their system crashed last week.

Hypotheses and parameters As in Exercise 1, for each of the following situations, define the parameter and write the null and alternative hypotheses in terms of parameter values. a) Seat-belt compliance in Massachusetts was 65% in 2008. The state wants to know if it has changed. b) Last year, a survey found that 45% of the employees were willing to pay for on-site day care. The company wants to know if that has changed. c) Regular card customers have a default rate of 6.7%. A credit card bank wants to know if that rate is different for their Gold card customers.

P-values Which of the following are true? If false, explain briefly. a) A very high P-value is strong evidence that the null hypothesis is false. b) A very low P-value proves that the null hypothesis is false. c) A high P-value shows that the null hypothesis is true. d) A P-value below 0.05 is always considered sufficient evidence to reject a null hypothesis.

More P-values Which of the following are true? If false, explain briefly. a) A very low P-value provides evidence against the null hypothesis. b) A high P-value is strong evidence in favor of the null hypothesis. c) A P-value above 0.10 shows that the null hypothesis is true. d) If the null hypothesis is true, you cant get a P-value below 0.01.

Hypotheses For each of the following, write out the null and alternative hypotheses, being sure to state whether the alternative is one-sided or two-sided. a) A company knows that last year 40% of its reports in accounting were on time. Using a random sample this year, it wants to see if that proportion has changed. b) Last year, 42% of the employees enrolled in at least one wellness class at the companys site. Using a survey, it wants to see whether a greater percentage is planning to take a wellness class this year. c) A political candidate wants to know from recent polls if shes going to garner a majority of votes in next weeks election.

More hypotheses For each of the following, write out the alternative hypothesis, being sure to indicate whether it is one-sided or two-sided. a) Consumer Reports discovered that 20% of a certain computer model had warranty problems over the first three months. From a random sample, the manufacturer wants to know if a new model has improved that rate. b) The last time a philanthropic agency requested donations, 4.75% of people responded. From a recent pilot mailing, they wonder if that rate has increased. c) A student wants to know if other students on her campus prefer Coke or Pepsi.

Errors For each of the following situations, state whether a Type I, a Type II, or neither error has been made. Explain briefly. a) A bank wants to see if the enrollment on their website is above 30% based on a small sample of customers. It tests H0: p = 0.3 vs. HA: p 7 0.3 and rejects the null hypothesis. Later the bank finds out that actually 28% of all customers enrolled. b) A student tests 100 students to determine whether other students on her campus prefer Coke or Pepsi and finds no evidence that preference for Coke is not 0.5. Later, a marketing company tests all students on campus and finds no difference. c) A pharmaceutical company tests whether a drug lifts the headache relief rate from the 25% achieved by the placebo. The test fails to reject the null hypothesis because the P-value is 0.465. Further testing shows that the drug actually relieves headaches in 38% of people.

More errors For each of the following situations, state whether a Type I, a Type II, or neither error has been made. a) A test of H0: p = 0.8 vs. HA: p 6 0.8 fails to reject the null hypothesis. Later it is discovered that p = 0.9. b) A test of H0: p = 0.5 vs. HA: p _ 0.5 rejects the null hypothesis. Later is it discovered that p = 0.65. c) A test of H0: p = 0.7 vs. HA: p 6 0.7 fails to reject the null hypothesis. Later is it discovered that p = 0.6.

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. The company runs 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.

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.

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.

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 studys P-value of 0.17 means in this context.

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.

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.

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.

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?

Groceries In January 2011, Yahoo surveyed 2400 U.S. men. 1224 of the men identified themselves as the primary grocery shopper in their household. a) Estimate the percentage of all American males who identify themselves as the primary grocery shopper. Use a 98% confidence interval. Check the conditions first. b) A grocery store owner believed that only 45% of men are the primary grocery shopper for their family, and targets his advertising accordingly. He wishes to conduct a hypothesis test to see if the fraction is in fact higher than 45%. What does your confidence interval indicate? Explain. c) What is the level of significance of this test? Explain.

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.

Approval In November 2011, Barack Obamas approval rating stood at 45% in Rasmussens daily tracking poll of 1500 randomly surveyed U.S. 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 Obamas approval rating was no worse than his November 2009 approval rating of 50%.

Hard times In June 2010, a random poll of 800 working men found that 9% had taken on a second job to help pay the bills. (www.careerbuilder.com) a) Estimate the true percentage of men that are taking on second jobs by constructing a 95% confidence interval. b) A pundit on a TV news show claimed that only 6% of working men had a second job. Use your confidence interval to test whether his claim is plausible given the poll data.

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.

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.

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 a for a hypothesis test? Explain. d) What impact does this change in the cutoff value have on the chance of each type of error?

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 a for a hypothesis test? Explain. d) What impact does this change in the cutoff value have on the chance of each type of error?

Second loan Exercise 23 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?

More spam Consider again the points-based spam filter described in Exercise 24. 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?

Homeowners In 2009, the U.S. Census Bureau reported that 67.4% 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?

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?

Testing cars A clean 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?

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?

Cars again As in Exercise 29, 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?

Production Consider again the task of the quality control inspectors in Exercise 30. 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?

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?

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.

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?

Ads A company 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 a = 0.05, 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.

Dropouts, part II Initially, 203 students signed up for the Stats course in Exercise 35. 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.

Testing the ads The company in Exercise 36 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.

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 a 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?

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 a for this test? d) What is the power of this test? (Hint: How many possibilities are in the alternative hypothesis?)

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 beforestill 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.

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.