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In a representative sample of adult Americans ages 26 to 32 years, \(27 \%\) indicated that they owned a fitness band that kept track of the number of steps walked each day and their daily activity levels ("Digital Democracy Survey", Deloitte Development LLC, 2016, www2, deloitte.com/us/en.html, retrieved November 30 , 2016). Suppose that the sample size was 500 . Is there convincing evidence that more than one-quarter of all adult Americans in this age group own a fitness band?

The article "Facebook Use and Academic Performance Among College Students" (Computers in Human Behavior [2015]: \(265-272\) ) estimated that \(87 \%\) percent of students at a large public university in California who are Facebook users update their status at least two times a day. This estimate was based on a random sample of 261 students at this university. a. Does this sample provide convincing evidence that more than \(80 \%\) of the students at this college who are Facebook users update their status at least two times a day? Test the relevant hypotheses using \(\alpha=0.05\). b. Would it be reasonable to generalize the conclusion from the test in Part (a) to all college students in the United States? Explain why or why not.

A number of initiatives on the topic of legalized gambling have appeared on state ballots. A political candidate has decided to support legalization of casino gambling if he is convinced that more than two-thirds of American adults approve of casino gambling. Suppose that 1035 of the people in a random sample of 1523 American adults said they approved of casino gambling. Is there convincing evidence that more than two-thirds approve?

Explain why failing to reject the null hypothesis in a hypothesis test does not mean there is convincing evidence that the null hypothesis is true.

Researchers at the University of Washington and Harvard University analyzed records of breast cancer screening and diagnostic evaluations ("Mammogram Cancer Scares More Frequent Than Thought," USA TODAY, April 16,1998 ). Discussing the benefits and downsides of the screening process, the article states that although the rate of falsepositives is higher than previously thought, if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall, but the rate of missed cancers would rise. Suppose that such a screening test is used to decide between a null hypothesis of \(H_{0}:\) no cancer is present and an alternative hypothesis of \(H_{a}:\) cancer is present. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) a. Would a false-positive (thinking that cancer is present when in fact it is not) be a Type I error or a Type II error? b. Describe a Type I error in the context of this problem, and discuss the consequences of making a Type I error. c. Describe a Type II error in the context of this problem, and discuss the consequences of making a Type II error. d. Recall the statement in the article that if radiologists were less aggressive in following up on suspicious tests, the rate of false-positives would fall but the rate of missed cancers would rise. What aspect of the relationship between the probability of a Type I error and the probability of a Type II error is being described here?

Refer to the instructions prior to Exercise \(10.90 .\) The paper "I Smoke but I Am Not a Smoker" ( Journal of American College Health [2010]: \(117-125\) ) describes a survey of 899 college students who were asked about their smoking behavior. Of the students surveyed, 268 classified themselves as nonsmokers, but said "yes" when asked later in the survey if they smoked. These students were classified as "phantom smokers" meaning that they did not view themselves as smokers even though they do smoke at times. The authors were interested in using these data to determine if there is convincing evidence that more than \(25 \%\) of college students fall into the phantom smoker category.

Past experience is that when individuals are approached with a request to fill out and return a particular questionnaire in a provided stamped and addressed envelope, the response rate is \(40 \%\). An investigator believes that if the person distributing the questionnaire were stigmatized in some obvious way, potential respondents would feel sorry for the distributor and thus tend to respond at a rate higher than \(40 \%\). To test this theory, a distributor wore an eye patch. Of the 200 questionnaires distributed by this individual, 109 were returned. Does this provide evidence that the response rate in this situation is greater than the previous rate of \(40 \%\) ? State and test the appropriate hypotheses using a significance plevel of 0.05 .