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In Exercises 21–24, identify the method of data collection the situation describes. A researcher compares incomes of people who live in rural areas with those who live in large urban areas.

THOUGHT PROVOKING You choose a random sample of 200 from a population of 2000 . Each person in the sample is asked how many hours of sleep he or she gets each night. The mean of your sample is 8 hours. Is it possible that the mean of the entire population is only \(7.5\) hours of sleep each night? Explain.

Suppose a random sample of size \(n\) is required to produce a margin of error of \(\pm E\). Write an expression in terms of \(n\) for the sample size needed to reduce the margin of error to \(\pm \frac{1}{2} E\). How many times must the sample size be increased to cut the margin of error in half? Explain.

In Exercises 21–24, identify the method of data collection the situation describes. A researcher places bacteria samples in two different climates. The researcher then measures the bacteria growth in each sample after 3 days.

In Exercises 25–28, explain why the survey question may be biased or otherwise introduce bias into the survey. Then describe a way to correct the flaw. A child asks, "Do you support the construction of a new children's hospital?"

In Exercises 29–32, determine whether the survey question may be biased or otherwise introduce bias into the survey. Explain your reasoning. "Do you think that renovating the old town hall would be a mistake?"

In Exercises 29–32, determine whether the survey question may be biased or otherwise introduce bias into the survey. Explain your reasoning. A police officer asks mall visitors, "Do you wear your seat belt regularly?"

An airline wants to know whether travelers have enough leg room on its planes. a. What method of data collection is appropriate for this situation? b. Describe a sampling method that is likely to give biased results. Explain. c. Describe a sampling method that is not likely to give biased results. Explain. d. Write one biased question and one unbiased question for this situation.

Simplify the expression. $$ \left(4^{3 / 2} \cdot 4^{1 / 4}\right)^4 $$