91Ó°ÊÓ

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. \(\mathrm{H}_{0}: p=0.5 \mathrm{vs} . \mathrm{H}_{\mathrm{A}}: p \neq 0.5\) a. A casino wants to know if their slot machine really delivers the 1 in 100 win rate that it claims. b. Last year, customers spent an average of \(\$ 35.32\) per visit to the company's website. Based on a random sample of purchases this year, the company wants to know if the mean this year has changed. c. A pharmaceutical company wonders if their new drug has a cure rate different from the \(30 \%\) reported by the placebo. d. 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.

A test preparation company claims that more than \(50 \%\) of the students who take their GRE prep course improve their scores by at least 10 points. a. Is the alternative to the null hypothesis more naturally one-sided or two- sided? Explain. b. A test run with randomly selected participants gives a P-value of 0.981 . What do you conclude? c. What would you have concluded if the \(\mathrm{P}\) -value had been \(0.019 ?\)

According to the 2010 Census, \(16 \%\) of the people in the United States are of Hispanic or Latino origin. One county supervisor believes her county has a different proportion of Hispanic people than the nation as a whole. She looks at their most recent survey data, which was a random sample of 437 county residents, and found that 44 of those surveyed are of Hispanic origin. a. State the hypotheses. b. Name the model and check appropriate conditions for a hypothesis test. c. Draw and label a sketch, and then calculate the test statistic and P-value. d. State your conclusion.

Empty houses According to the 2010 Census, \(11.4 \%\) of all housing units in the United States were vacant. A county supervisor wonders if her county is different from this. She randomly selects 850 housing units in her county and finds that 129 of the housing units are vacant. a. State the hypotheses. b. Name the model and check appropriate conditions for a hypothesis test. c. Draw and label a sketch, and then calculate the test statistic and P-value. d. State your conclusion.

Marriage In \(1960,\) census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting longer to get married. We want to find out if the mean age of first marriage has increased since then. a. Write appropriate hypotheses. b. We plan to test our hypothesis by selecting a random sample of 40 men who married for the first time last year. Do you think the necessary assumptions for inference are satisfied? Explain. c. Describe the approximate sampling distribution model for the mean age in such samples. d. The men in our sample married at an average age of 24.2 years, with a standard deviation of 5.3 years.

Expensive medicine Developing a new drug can be an expensive process, resulting in high costs to patients. A pharmaceutical company has developed a new drug to reduce cholesterol, and it will conduct a clinical trial to compare the effectiveness to the most widely used current treatment. The results will be analyzed using a hypothesis test. a. If the test yields a low P-value and the researcher rejects the null hypothesis that the new drug is not more effective, but it actually is not better, what are the consequences of such an error? b. If the test yields a high \(\mathrm{P}\) -value and the researcher fails to reject the null hypothesis, but the new drug is more effective, what are the consequences of such an error?

Write the null and alternative hypotheses you would use to test each of the following situations: a. A governor is concerned about his "negatives"- -the percentage of state residents who express disapproval of his job performance. His political committee pays for a series of TV ads, hoping that they can keep the negatives below \(30 \%\). They will use follow-up polling to assess the ads' effectiveness. b. Is a coin fair? c. Only about \(20 \%\) of people who try to quit smoking succeed. Sellers of a motivational tape claim that listening to the recorded messages can help people quit.

More hypotheses Write the null and alternative hypotheses you would use to test each situation. a. In the \(1950 \mathrm{~s}\), only about \(40 \%\) of high school graduates went on to college. Has the percentage changed? b. Twenty percent of cars of a certain model have needed costly transmission work after being driven between 50,000 and 100,000 miles. The manufacturer hopes that a redesign of a transmission component has solved this problem. c. We field-test a new-flavor soft drink, planning to market it only if we are sure that over \(60 \%\) of the

Dice The seller of a loaded die claims that it will favor the outcome \(6 .\) We don't believe that claim, and roll the die 200 times to test an appropriate hypothesis. Our P-value turns out to be 0.03. Which conclusion is appropriate? Explain. a. There's a \(3 \%\) chance that the die is fair. \(\mathrm{b}\). There's a \(97 \%\) chance that the die is fair. c. There's a \(3 \%\) chance that a loaded die could randomly produce the results we observed, so it's reasonable to conclude that the die is fair. d. There's a \(3 \%\) chance that a fair die could randomly produce the results we observed, so it's reasonable to conclude that the die is loaded.

Relief A company's old antacid formula provided relief for \(70 \%\) of the people who used it. The company tests a new formula to see if it is better and gets a P-value of \(0.27 .\) Is it reasonable to conclude that the new formula and the old one are equally effective? Explain.