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A Vermont study published by the American Academy of Pediatrics examined parental influence on teenagers' decisions to smoke. A group of students who had never smoked were questioned about their parents' attitudes toward smoking. These students were questioned again two years later to see if they had started smoking. The researchers found that, among the 284 students who indicated that their parents disapproved of kids smoking, 54 had become established smokers. Among the 41 students who initially said their parents were lenient about smoking, 11 became smokers. Do these data provide strong evidence that parental attitude influences teenagers' decisions about smoking? a. What kind of design did the researchers use? b. Write appropriate hypotheses. c. Are the assumptions and conditions necessary for inference satisfied? d. Test the hypothesis and state your conclusion. e. Explain in this context what your P-value means. \(\mathrm{f}\). If it is later found that parental attitudes actually do influence teens' decisions to smoke, which type of

The Journal of the American Medical Association reported a study examining the possible impact of air pollution caused by the \(9 / 11\) attack on New York's World Trade Center on the weight of babies. Researchers found that \(8 \%\) of 182 babies born to mothers who were exposed to heavy doses of soot and ash on September 11 were classified as having low birthweight. Only \(4 \%\) of 2300 babies born in another New York City hospital whose mothers had not been near the site of the disaster were similarly classified. Does this indicate a possibility that air pollution might be linked to a significantly higher proportion of low-weight babies? a. Test an appropriate hypothesis at \(\alpha=0.10\) and state your conclusion. b. If you concluded there is a difference, estimate that difference with a confidence interval and interpret that interval in context.

One month before the election, a poll of 630 randomly selected voters showed \(54 \%\) planning to vote for a certain candidate. A week later, it became known that he had had an extramarital affair, and a new poll showed only \(51 \%\) of 1010 voters supporting him. Do these results indicate a decrease in voter support for his candidacy? a. Test an appropriate hypothesis and state your conclusion. b. If you concluded there was a difference, estimate that difference with a confidence interval and interpret your interval in context.

Researchers comparing the effectiveness of two pain medications randomly selected a group of patients who had been complaining of a certain kind of joint pain. They randomly divided these people into two groups, then administered the pain killers. Of the 112 people in the group who received medication A, 84 said this pain reliever was effective. Of the 108 people in the other group, 66 reported that pain reliever B was effective. a. Write a \(95 \%\) confidence interval for the percent of people who may get relief from this kind of joint pain by using medication A. Interpret your interval. b. Write a \(95 \%\) confidence interval for the percent of people who may get relief by using medication B. Interpret your interval. c. Do the intervals for \(A\) and B overlap? What do you think this means about the comparative effectiveness of these medications? d. Find a \(95 \%\) confidence interval for the difference in the proportions of people who may find these medications effective. Interpret your interval. e. Does this interval contain zero? What does that mean? f. Why do the results in parts \(c\) and e seem contradictory? If we want to compare the effectiveness of these two pain relievers, which is the correct approach? Why?

Candidates for political office realize that different levels of support among men and women may be a crucial factor in determining the outcome of an election. One candidate finds that \(52 \%\) of 473 men polled say they will vote for him, but only \(45 \%\) of the 522 women in the poll express support. a. Write a \(95 \%\) confidence interval for the percent of male voters who may vote for this candidate. Interpret your interval. b. Write a \(95 \%\) confidence interval for the percent of female voters who may vote for him. Interpret your interval. c. Do the intervals for males and females overlap? What do you think this means about the gender gap? d. Find a \(95 \%\) confidence interval for the difference in the proportions of males and females who will vote for this candidate. Interpret your interval. e. Does this interval contain zero? What does that mean? f. Why do the results in parts \(c\) and e seem contradictory? If we want to see if there is a gender gap among voters with respect to this candidate, which is the correct approach? Why?

In the July 2007 issue, Consumer Reports examined the calorie content of two kinds of hot dogs: meat (usually a mixture of pork, turkey, and chicken) and all beef. The researchers purchased samples of several different brands. The meat hot dogs averaged 111.7 calories, compared to 135.4 for the beef hot dogs. A test of the null hypothesis that there's no difference in mean calorie content yields a P-value of 0.124 . Would a \(95 \%\) confidence interval for \(\mu_{M e a t}-\mu_{B e e f}\) include \(0 ?\) Explain.

The Consumer Reports article described in Exercise 51 also listed the fat content (in grams) for samples of beef and meat hot dogs. The resulting \(90 \%\) confidence interval for \(\mu_{M e a t}-\mu_{B e e f}\) is (-6.5,-1.4) a. The endpoints of this confidence interval are negative numbers. What does that indicate? b. What does the fact that the confidence interval does not contain 0 indicate? c. If we use this confidence interval to test the hypothesis that \(\mu_{M e a t}-\mu_{B e e f}=0\) what's the corresponding alpha level?

In Exercise 53 , we saw a \(90 \%\) confidence interval of (-6.5,-1.4) grams for \(\mu_{\text {Meat }}-\mu_{\text {Beef }}\) the difference in mean fat content for meat vs. all-beef hot dogs. Explain why you think each of the following statements is true or false: a. If I eat a meat hot dog instead of a beef dog, there's a \(90 \%\) chance I'll consume less fat. b. \(90 \%\) of meat hot dogs have between 1.4 and 6.5 grams less fat than a beef hot dog. c. I'm \(90 \%\) confident that meat hot dogs average between 1.4 and 6.5 grams less fat than the beef hot dogs. d. If I were to get more samples of both kinds of hot dogs, \(90 \%\) of the time the meat hot dogs would average between 1.4 and 6.5 grams less fat than the beef hot dogs. e. If I tested more samples, l'd expect about \(90 \%\) of the resulting confidence intervals to include the true difference in mean fat content between the two kinds of hot dogs.

The Core Plus Mathematics Project (CPMP) is an innovative approach to teaching Mathematics that engages students in group investigations and mathematical modeling. After field tests in 36 high schools over a three-year period, researchers compared the performances of CPMP students with those taught using a traditional curriculum. In one test, students had to solve applied algebra problems using calculators. Scores for 320 CPMP students were compared to those of a control group of 273 students in a traditional math program. Computer software was used to create a confidence interval for the difference in mean scores. (Journal for Research in Mathematics Education, 31, no. 3) Conf level: \(95 \%\) Variable: Mu(CPMP) - Mu(CtrI) Interval: (5.573,11.427) a. What's the margin of error for this confidence interval? b. If we had created a \(98 \% \mathrm{Cl}\), would the margin of error be larger or smaller? c. Explain what the calculated interval means in this context. d. Does this result suggest that students who learn mathematics with CPMP will have significantly higher mean scores in algebra than those in traditional programs? Explain.

A man who moves to a new city sees that there are two routes he could take to work. A neighbor who has lived there a long time tells him Route A will average 5 minutes faster than Route B. The man decides to experiment. Each day, he flips a coin to determine which way to go, driving each route 20 days. He finds that Route A takes an average of 40 minutes, with standard deviation 3 minutes, and Route B takes an average of 43 minutes, with standard deviation 2 minutes. Histograms of travel times for the routes are roughly symmetric and show no outliers. a. Find a \(95 \%\) confidence interval for the difference in average commuting time for the two routes. (From technology, \(d f=33.1 .)\) b. Should the man believe the old-timer's claim that he can save an average of 5 minutes a day by always driving Route A? Explain.