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How Often Do You Use Cash? In a survey \(^{13}\) of 1000 American adults conducted in April 2012 . \(43 \%\) reported having gone through an entire week without paying for anything in cash. Test to see if this sample provides evidence that the proportion of all American adults going a week without paying cash is greater than \(40 \%\). Use the fact that a randomization distribution is approximately normally distributed with a standard error of \(S E=0.016 .\) Show all details of the test and use a \(5 \%\) significance level.

Incentives for Quitting Smoking: Group or Individual? In a smoking cessation program, over 2000 smokers who were trying to quit were randomly assigned to either a group program or an individual program. After six months in the program, 148 of the 1080 in the group program were successfully abstaining from smoking. while 120 of the 990 in the individual program were successful. \({ }^{15}\) We wish to test to see if this data provide evidence of a difference in the proportion able to quit smoking between smokers in a group program and smokers in an individual program. (a) State the null and alternative hypotheses, and give the notation and value of the sample statistic. (b) Use a randomization distribution and the observed sample statistic to find the p-value. (c) Give the mean and standard error of the normal distribution that most closely matches the randomization distribution, and then use this normal distribution with the observed sample statistic to find the p-value. (d) Use the standard error found from the randomization distribution in part (b) to find the standardized test statistic, and then use that test statistic to find the p-value using a standard normal distribution. (e) Compare the p-values from parts (b), (c), and (d). Use any of these p-values to give the conclusion of the test.

Incentives for Quitting Smoking: Do They Work? Exercise 5.31 describes a study examining incentives to quit smoking. With no incentives, the proportion of smokers trying to quit who are still abstaining six months later is about 0.06 . Participants in the study were randomly assigned to one of four different incentives, and the proportion successful was measured six months later. Of the 498 participants in the group with the least success, 47 were still abstaining from smoking six months later. We wish to test to see if this provides evidence that even the smallest incentive works better than the proportion of 0.06 with no incentive at all. (a) State the null and alternative hypotheses, and give the notation and value of the sample statistic. (b) Use a randomization distribution and the observed sample statistic to find the p-value. (c) Give the mean and standard error of the normal distribution that most closely matches the randomization distribution, and then use this normal distribution with the observed sample statistic to find the p-value. (d) Use the standard error found from the randomization distribution in part (b) to find the standardized test statistic, and then use that test statistic to find the p-value using a standard normal distribution. (e) Compare the p-values from parts (b), (c), and (d). Use any of these p-values to give the conclusion of the test.

Find the \(z^{*}\) values based on a standard normal distribution for each of the following. (a) An \(80 \%\) confidence interval for a proportion. (b) An \(84 \%\) confidence interval for a slope. (c) A \(92 \%\) confidence interval for a standard deviation.

Find the \(z^{*}\) values based on a standard normal distribution for each of the following. (a) An \(86 \%\) confidence interval for a correlation. (b) A \(94 \%\) confidence interval for a difference in proportions. (c) A \(96 \%\) confidence interval for a proportion.

Find the indicated confidence interval. Assume the standard error comes from a bootstrap distribution that is approximately normally distributed. A \(95 \%\) confidence interval for a proportion \(p\) if the sample has \(n=100\) with \(\hat{p}=0.43,\) and the standard error is \(S E=0.05\).

Find the indicated confidence interval. Assume the standard error comes from a bootstrap distribution that is approximately normally distributed. A \(95 \%\) confidence interval for a mean \(\mu\) if the sample has \(n=50\) with \(\bar{x}=72\) and \(s=12,\) and the standard error is \(S E=1.70 .\)

Hearing Loss in Teenagers A recent study \(^{20}\) found that, of the 1771 participants aged 12 to 19 in the National Health and Nutrition Examination Survey, \(19.5 \%\) had some hearing loss (defined as a loss of 15 decibels in at least one ear). This is a dramatic increase from a decade ago. The sample size is large enough to use the normal distribution, and a bootstrap distribution shows that the standard error for the proportion is \(S E=0.009 .\) Find and interpret a \(90 \%\) confidence interval for the proportion of teenagers with some hearing loss.

Where Is the Best Seat on the Plane? A survey of 1000 air travelers \(^{21}\) found that \(60 \%\) prefer a window seat. The sample size is large enough to use the normal distribution, and a bootstrap distribution shows that the standard error is \(S E=0.015 .\) Use a normal distribution to find and interpret a \(99 \%\) confidence interval for the proportion of air travelers who prefer a window seat.

Smoke-Free Legislation and Asthma Hospital admissions for asthma in children younger than 15 years was studied \(^{22}\) in Scotland both before and after comprehensive smoke-free legislation was passed in March \(2006 .\) Monthly records were kept of the annualized percent change in asthma admissions. For the sample studied, before the legislation, admissions for asthma were increasing at a mean rate of \(5.2 \%\) per year. The standard error for this estimate is \(0.7 \%\) per year. After the legislation, admissions were decreasing at a mean rate of \(18.2 \%\) per year, with a standard error for this mean of \(1.79 \% .\) In both cases, the sample size is large enough to use a normal distribution. (a) Find and interpret a \(95 \%\) confidence interval for the mean annual percent rate of change in childhood asthma hospital admissions in Scotland before the smoke-free legislation. (b) Find a \(95 \%\) confidence interval for the same quantity after the legislation. (c) Is this an experiment or an observational study? (d) The evidence is quite compelling. Can we conclude cause and effect?