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About \(77 \%\) of young adults think they can achieve the American dream. Determine if the following statements are true or false, and explain your reasoning. \({ }^{13}\) (a) The distribution of sample proportions of young Americans who think they can achieve the American dream in samples of size 20 is left skewed. (b) The distribution of sample proportions of young Americans who think they can achieve the American dream in random samples of size 40 is approximately normal since \(n \geq 30\). (c) A random sample of 60 young Americans where \(85 \%\) think they can achieve the American dream would be considered unusual. (d) A random sample of 120 young Americans where \(85 \%\) think they can achieve the American dream would be considered unusual.

About \(25 \%\) of young Americans have delayed starting a family due to the continued economic slump. Determine if the following statements are true or false, and explain your reasoning. \(^{14}\) (a) The distribution of sample proportions of young Americans who have delayed starting a family due to the continued economic slump in random samples of size 12 is right skewed. (b) In order for the distribution of sample proportions of young Americans who have delayed starting a family due to the continued economic slump to be approximately normal, we need random samples where the sample size is at least 40 . (c) A random sample of 50 young Americans where \(20 \%\) have delayed starting a family due to the continued economic slump would be considered unusual. (d) A random sample of 150 young Americans where \(20 \%\) have delayed starting a family due to the continued economic slump would be considered unusual. (e) Tripling the sample size will reduce the standard error of the sample proportion by one-third.

A Kaiser Family Foundation poll for US adults in 2019 found that \(79 \%\) of Democrats, \(55 \%\) of Independents, and \(24 \%\) of Republicans supported a generic "National Health Plan". There were 347 Democrats, 298 Republicans, and 617 Independents surveyed. (a) A political pundit on TV claims that a majority of Independents support a National Health Plan. Do these data provide strong evidence to support this type of statement? (b) Would you expect a confidence interval for the proportion of Independents who oppose the public option plan to include \(0.5 ?\) Explain.

Exercise 6.12 presents the results of a poll where \(48 \%\) of 331 Americans who decide to not go to college do so because they cannot afford it. (a) Calculate a \(90 \%\) confidence interval for the proportion of Americans who decide to not go to college because they cannot afford it, and interpret the interval in context. (b) Suppose we wanted the margin of error for the \(90 \%\) confidence level to be about \(1.5 \% .\) How large of a survey would you recommend?

Exercise 6.11 presents the results of a poll evaluating support for a generic "National Health Plan" in the US in 2019 , reporting that \(55 \%\) of Independents are supportive. If we wanted to estimate this number to within \(1 \%\) with \(90 \%\) confidence, what would be an appropriate sample size?

In July 2008 the US National Institutes of Health announced that it was stopping a clinical study early because of unexpected results. The study population consisted of HIV-infected women in sub-Saharan Africa who had been given single dose Nevaripine (a treatment for HIV) while giving birth, to prevent transmission of HIV to the infant. The study was a randomized comparison of continued treatment of a woman (after successful childbirth) with Nevaripine vs Lopinavir, a second drug used to treat HIV. 240 women participated in the study; 120 were randomized to each of the two treatments. Twentyfour weeks after starting the study treatment, each woman was tested to determine if the HIV infection was becoming worse (an outcome called virologic failure). Twenty-six of the 120 women treated with Nevaripine experienced virologic failure, while 10 of the 120 women treated with the other drug experienced virologic failure. \({ }^{36}\) (a) Create a two-way table presenting the results of this study. (b) State appropriate hypotheses to test for difference in virologic failure rates between treatment groups. (c) Complete the hypothesis test and state an appropriate conclusion.