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A survey is planned to estimate the proportion of voters who support a proposed gun control law. The estimate should be within a margin of error of \(\pm 2 \%\) with \(95 \%\) confidence, and we do not have any prior knowledge about the proportion who might support the law. How many people need to be included in the sample?

When we want \(95 \%\) confidence and use the conservative estimate of \(p=0.5,\) we can use the simple formula \(n=1 /(M E)^{2}\) to estimate the sample size needed for a given margin of error ME. In Exercises 6.40 to 6.43, use this formula to determine the sample size needed for the given margin of error. A margin of error of 0.01

When we want \(95 \%\) confidence and use the conservative estimate of \(p=0.5,\) we can use the simple formula \(n=1 /(M E)^{2}\) to estimate the sample size needed for a given margin of error ME. In Exercises 6.40 to 6.43, use this formula to determine the sample size needed for the given margin of error. A margin of error of 0.04 .

Antibiotics in Infancy Exercise 2.19 describes a Canadian longitudinal study that examines whether giving antibiotics in infancy increases the likelihood that the child will be overweight later in life. The study included 616 children and found that 438 of the children had received antibiotics during the first year of life. Test to see if this provides evidence that more than \(70 \%\) of Canadian children receive antibiotics during the first year of life. Show all details of the hypothesis test, including hypotheses, the standardized test statistic, the p-value, the generic conclusion using a \(5 \%\) significance level, and a conclusion in context.

Left-Handed Lawyers Approximately \(10 \%\) of Americans are left-handed (we will treat this as a known population parameter). A study on the relationship between handedness and profession found that in a random sample of 105 lawyers, 16 of them were left-handed. \({ }^{13}\) Test the hypothesis that the proportion of left-handed lawyers differs from the proportion of left-handed Americans. (a) Clearly state the null and alternative hypotheses. (b) Calculate the test statistic and p-value. (c) What do we conclude at the \(5 \%\) significance level? At the \(10 \%\) significance level?

Home Field Advantage in Baseball 2009 There were 2430 Major League Baseball (MLB) games played in \(2009,\) and the home team won the game in \(54.9 \%\) of the games. \({ }^{15}\) If we consider the games played in 2009 as a sample of all MLB games, test to see if there is evidence, at the \(1 \%\) level, that the home team wins more than half the games. Show all details of the test.

Do You Know Your Neighbors? A survey of 2255 randomly selected US adults found that \(51 \%\) said they know all or most of their neighbors. \({ }^{17}\) Does this provide evidence that more than half of US adults know most or all of their neighbors?

Is B a Good Choice on a Multiple-Choice Exam? Multiple-choice questions on Advanced Placement exams have five options: \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\) and \(\mathrm{E}\). A random sample of the correct choice on 400 multiple-choice questions on a variety of \(\mathrm{AP}\) exams \(^{18}\) shows that \(\mathrm{B}\) was the most common correct choice, with 90 of the 400 questions having \(\underline{B}\) as the answer. Does this provide evidence that \(\mathrm{B}\) is more likely to be the correct choice than would be expected if all five options were equally likely? Show all details of the test. The data are available in APMultipleChoice.

Do Babies Understand Probability? Can babies reason probabilistically? A study \(^{19}\) investigates this by showing ten- to twelve-month-old infants two jars of lollipop-shaped objects colored pink or black. Each infant first crawled or walked to whichever color they wanted, determining their "preferred" color. They were then given the choice between two jars that had the same number of preferred objects, but that differed in their probability of getting the preferred color; each jar had 12 in the preferred color and either 4 or 36 in the other color. Babies choosing randomly or based on the absolute number of their preferred color would choose equally between the two jars, while babies understanding probability would more often choose the jar with the higher proportion of their preferred color. Of the 24 infants studied, 18 chose the jar with the higher proportion of their preferred color. Are infants more likely to choose the jar with the higher proportion of their preferred color? (a) State the null and alternative hypotheses. (b) Give the relevant sample statistic, using correct notation. (c) Which of the following should be used to calculate a p-value for this dataset? A randomization test, a test using the normal distribution, or either one? Why? (d) Find a p-value using a method appropriate for this data situation. (e) Make a conclusion in context, using \(\alpha=0.05\).

If random samples of the given size are drawn from a population with the given mean and standard deviation, find the standard error of the distribution of sample means. Samples of size 1000 from a population with mean 28 and standard deviation 5