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Three different statistics are being considered for estimating a population characteristic. The sampling distributions of the three statistics are shown in the following illustration: Which statistic would you recommend? Explain your choice.

For estimating a population characteristic, why is an unbiased statistic generally preferred over a biased statistic? Does unbiasedness alone guarantee that the estimate will be close to the actual value of the population characteristic? Explain.

A researcher wants to estimate the proportion of students enrolled at a university who are registered to vote. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion was \(p=0.4\) or \(p=0.8 ?\)

If two statistics are available for estimating a population characteristic, under what circumstances might you choose a biased statistic over an unbiased statistic?

Consider taking a random sample from a population with \(p=0.40\). a. What is the standard error of \(\hat{p}\) for random samples of size \(100 ?\) b. Would the standard error of \(\hat{p}\) be greater for samples of size 100 or samples of size \(200 ?\) c. If the sample size were doubled from 100 to 200 , by what factor would the standard error of \(\hat{p}\) decrease?

The paper "Sleeping with Technology: Cognitive, Affective and Technology Usage Predictors of Sleep Problems Among College Students" (Sleep Health [2016]: 49-56) summarized data from a survey of a sample of college students. Of the 734 students surveyed, 125 reported that they sleep with their cell phones near the bed and check their phones for something other than the time at least twice during the night. For purposes of this exercise, assume that this sample is representative of college students in the United States. a. Use the given information to estimate the proportion of college students who check their cell phones for something other than the time at least twice during the night. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.

The report "The 2016 Consumer Financial Literacy Survey" (The National Foundation for Credit Counseling, www .nfcc.org, retrieved October 28,2016 ) summarized data from a representative sample of 1668 adult Americans. When asked if they typically carry credit card debt from month to month, 584 of these people responded "yes." a. Use the given information to estimate the proportion of adult Americans who carry credit card debt from month to month. b. Verify that the conditions needed in order for the margin of error formula to be appropriate are met. c. Calculate the margin of error. d. Interpret the margin of error in the context of this problem.

Business Insider reported that a study commissioned by eBay Motors found that nearly \(40 \%\) of millennials who drive a car that is more than 5 years old have named their cars ("Millennials Have an Odd Habit When It Comes to Their Cars," April 14,2016 ). a. Assuming that the sample was selected to be representative of the population of millennials who drive a car that is more than 5 years old, what is an estimate of the population proportion who have named their car? b. Suppose that the sample size for the study described was 800. Calculate and interpret the margin of error associated with your estimate in Part (a).

Suppose that a city planning commission wants to know the proportion of city residents who support installing streetlights in the downtown area. Two different people independently selected random samples of city residents and used their sample data to construct the following confidence intervals for the population proportion: Interval 1:(0.28,0.34) Interval 2:(0.31,0.33) (Hint: Consider the formula for the confidence interval given on page 444.) a. Explain how it is possible that the two confidence intervals are not centered in the same place. b. Which of the two intervals conveys more precise information about the value of the population proportion? c. If both confidence intervals have a \(95 \%\) confidence level, which confidence interval was based on the smaller sample size? How can you tell? d. If both confidence intervals were based on the same sample size, which interval has the higher confidence level? How can you tell?

For each of the following choices, explain which one would result in a wider large-sample confidence interval for \(p:\) a. \(90 \%\) confidence level or \(95 \%\) confidence level b. \(n=100\) or \(n=400\)