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Medical personnel are required to report suspected cases of child abuse. Because some diseases have symptoms that are similar to those of child abuse, doctors who see a child with these symptoms must decide between two competing hypotheses: \(H_{0}:\) symptoms are due to child abuse \(H:\) symptoms are not due to child abuse (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) The article "Blurred Line Between Illness, Abuse Creates Problem for Authorities" (Macon Telegraph, February \(28,\) 2000 ) included the following quote from a doctor in Atlanta regarding the consequences of making an incorrect decision: "If it's disease, the worst you have is an angry family. If it is abuse, the other kids (in the family) are in deadly danger." a. For the given hypotheses, describe Type I and Type II errors. b. Based on the quote regarding consequences of the two kinds of error, which type of error is considered more serious by the doctor quoted? Explain.

How accurate are DNA paternity tests? By comparing the DNA of the baby and the DNA of a man that is being tested, one maker of DNA paternity tests claims that their test is \(100 \%\) accurate if the man is not the father and \(99.99 \%\) accurate if the man is the father (IDENTIGENE, www.dnatesting .com/paternity-test-questions/paternity-test-accuracy/, retrieved November 16,2016 ). a. Consider using the result of this DNA paternity test to decide between the following two hypotheses: \(H_{0}:\) a particular man is not the father \(H:\) a particular man is the father In the context of this problem, describe Type I and Type II errors. (Although these are not hypotheses about a population characteristic, this exercise illustrates the definitions of Type I and Type II errors.) b. Based on the information given, what are the values of \(\alpha\), the probability of a Type I error, and \(\beta\), the probability of a Type II error?

In the report "Healthy People 2020 Objectives for the Nation," The Centers for Disease Control and Prevention (CDC) set a goal of 0.341 for the proportion of mothers who will still be breastfeeding their babies one year after birth (www.cdc.gov/breastfeeding/policy /hp2020.htm, April 11, 2016, retrieved November 28, 2016). The CDC also estimated the proportion who were still being breastfed one year after birth to be 0.307 for babies born in 2013 (www.cdc.gov/breastfeeding /pdf/2016breastfeedingreportcard.pdf, retrieved November 28,2016) . This estimate was based on a survey of women who had given birth in 2013 . Suppose that the survey used a random sample of 1000 mothers and that you want to use the survey data to decide if there is evidence that the goal is not being met. Let \(p\) denote the population proportion of all mothers of babies born in 2013 who were still breast-feeding at 12 months. (Hint: See Example \(10.10 .)\) a. Describe the shape, center, and variability of the sampling distribution of \(\hat{p}\) for random samples of size 1000 if the null hypothesis \(H_{0}: p=0.341\) is true. b. Would you be surprised to observe a sample proportion as small as \(\hat{p}=0.333\) for a sample of size 1000 if the null hypothesis \(H_{0}: p=0.341\) were true? Explain why or why not. c. Would you be surprised to observe a sample proportion as small as \(\hat{p}=0.310\) for a sample of size 1000 if the null hypothesis \(H_{0}: p=0.341\) were true? Explain why or why not. d. The actual sample proportion observed in the study was \(p=0.307\). Based on this sample proportion, is there convincing evidence that the goal is not being met, or is the observed sample proportion consistent with what you would expect to see when the null hypothesis is true? Support your answer with a probability calculation.

Use the definition of the \(P\) -value to explain the following: a. Why \(H_{0}\) would be rejected if \(P\) -value \(=0.003\) b. Why \(H_{0}\) would not be rejected if \(P\) -value \(=0.350\)

For which of the following \(P\) -values will the null hypothesis be rejected when performing a test with a significance level of \(0.05 ?\) a. 0.001 d. 0.047 b. 0.021 e. 0.148 c. 0.078

Explain why a \(P\) -value of 0.002 would be interpreted as strong evidence against the null hypothesis.

The paper "Teens and Distracted Driving"" (Pew Internet \& American Life Project, 2009 ) reported that in a representative sample of 283 American teens age 16 to \(17,\) there were 74 who indicated that they had sent a text message while driving. For purposes of this exercise, assume that this sample is a random sample of 16- to 17 -year-old Americans. Do these data provide convincing evidence that more than a quarter of Americans age 16 to 17 have sent a text message while driving? Test the appropriate hypotheses using a significance level of 0.01 . (Hint: See Example 10.11 .)

Refer to the instructions given prior to Exercise \(10.57 .\) The paper "Pathological Video-Game Use Among Youth Ages 8 to 18: A National Study" (Psychological Science [2009]: \(594-601\) ) summarizes data from a random sample of 1178 students age 8 to \(18 .\) The paper reported that for the students in the sample, the mean amount of time spent playing video games was 13.2 hours per week. The researchers were interested in using the data to estimate the mean amount of time spent playing video games for students age 8 to 18 .

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample \(z\) test appropriate? a. \(H_{0}: p=0.8, n=40\) b. \(H_{0}: p=0.4, n=100\) c. \(H_{0}: p=0.1, n=50\) d. \(H_{0}: p=0.05, n=750\)

In a survey of 1000 women age 22 to 35 who work full-time, 540 indicated that they would be willing to give up some personal time in order to make more money (USA TODAY, March 4,2010 ). The sample was selected to be representative of women in the targeted age group. a. Do the sample data provide convincing evidence that a majority of women age 22 to 35 who work fulltime would be willing to give up some personal time for more money? Test the relevant hypotheses using \(\alpha=0.01\) b. Would it be reasonable to generalize the conclusion from Part (a) to all working women? Explain why or why not.