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A nonprofit wants to understand the fraction of households that have elevated levels of lead in their drinking water. They expect at least \(5 \%\) of homes will have elevated levels of lead, but not more than about \(30 \%\). They randomly sample 800 homes and work with the owners to retrieve water samples, and they compute the fraction of these homes with elevated lead levels. They repeat this 1,000 times and build a distribution of sample proportions. (a) What is this distribution called? (b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning. (c) If the proportions are distributed around \(8 \%\), what is the variability of the distribution? (d) What is the formal name of the value you computed in (c)? (e) Suppose the researchers' budget is reduced, and they are only able to collect 250 observations per sample, but they can still collect 1,000 samples. They build a new distribution of sample proportions. How will the variability of this new distribution compare to the variability of the distribution when each sample contained 800 observations?

Of all freshman at a large college, \(16 \%\) made the dean's list in the current year. As part of a class project, students randomly sample 40 students and check if those students made the list. They repeat this 1,000 times and build a distribution of sample proportions. (a) What is this distribution called? (b) Would you expect the shape of this distribution to be symmetric, right skewed, or left skewed? Explain your reasoning. (c) Calculate the variability of this distribution. (d) What is the formal name of the value you computed in (c)? (e) Suppose the students decide to sample again, this time collecting 90 students per sample, and they again collect 1,000 samples. They build a new distribution of sample proportions. How will the variability of this new distribution compare to the variability of the distribution when each sample contained 40 observations?

A poll conducted in 2013 found that \(52 \%\) of U.S. adult Twitter users get at least some news on Twitter. \(^{13}\). The standard error for this estimate was \(2.4 \%\), and a normal distribution may be used to model the sample proportion. Construct a \(99 \%\) confidence interval for the fraction of U.S. adult Twitter users who get some news on Twitter, and interpret the confidence interval in context.

A study suggests that \(60 \%\) of college student spend 10 or more hours per week communicating with others online. You believe that this is incorrect and decide to collect your own sample for a hypothesis test. You randomly sample 160 students from your dorm and find that \(70 \%\) spent 10 or more hours a week communicating with others online. A friend of yours, who offers to help you with the hypothesis test, comes up with the following set of hypotheses. Indicate any errors you see. $$ \begin{array}{l} H_{0}: \hat{p}<0.6 \\ H_{A}: \hat{p}>0.7 \end{array} $$

Exercise 5.11 provides a \(95 \%\) confidence interval for the mean waiting time at an emergency room (ER) of (128 minutes, 147 minutes). Answer the following questions based on this interval. (a) A local newspaper claims that the average waiting time at this ER exceeds 3 hours. Is this claim supported by the confidence interval? Explain your reasoning. (b) The Dean of Medicine at this hospital claims the average wait time is 2.2 hours. Is this claim supported by the confidence interval? Explain your reasoning. (c) Without actually calculating the interval, determine if the claim of the Dean from part (b) would be supported based on a \(99 \%\) confidence interval?

In each part below, there is a value of interest and two scenarios (I and II). For each part, report if the value of interest is larger under scenario I, scenario II, or whether the value is equal under the scenarios. (a) The standard error of \(\hat{p}\) when (I) \(n=125\) or (II) \(n=500\). (b) The margin of error of a confidence interval when the confidence level is (I) \(90 \%\) or (II) \(80 \%\). (c) The p-value for a Z-statistic of 2.5 calculated based on a (I) sample with \(n=500\) or based on a (II) sample with \(n=1000\). (d) The probability of making a Type 2 Error when the alternative hypothesis is true and the significance level is (I) 0.05 or (II) 0.10 .

In Exercise \(5.19,\) we learned that a Rasmussen Reports survey of \(1,000 \mathrm{US}\) adults found that \(42 \%\) believe raising the minimum wage will help the economy. Construct a \(99 \%\) confidence interval for the true proportion of US adults who believe this.

It is believed that nearsightedness affects about \(8 \%\) of all children. In a random sample of 194 children, 21 are nearsighted. Conduct a hypothesis test for the following question: do these data provide evidence that the \(8 \%\) value is inaccurate?

Determine whether the following statement is true or false, and explain your reasoning: "With large sample sizes, even small differences between the null value and the observed point estimate can be statistically significant."

Suppose you conduct a hypothesis test based on a sample where the sample size is \(n=50,\) and arrive at a p-value of 0.08 . You then refer back to your notes and discover that you made a careless mistake, the sample size should have been \(n=500\). Will your p-value increase, decrease, or stay the same? Explain.