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Chapter 8: Problem 29
What are the parameters of a normal distribution and a \(t\) distribution? Explain.
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When calculating a confidence interval for the population mean \(\mu\) with a known population standard deviation \(\sigma\), describe the effects of the following two changes on the confidence interval: (1) doubling the sample size, (2) quadrupling (multiplying by 4) the sample size. Give two reasons why this relationship does not hold true if you are calculating a confidence interval for the population mean \(\mu\) with an unknown population standard deviation.
The mean time taken to design a house plan by 40 architects was found to be 23 hours with a standard deviation of \(3.75\) hours. a. Construct a \(98 \%\) confidence interval for the population mean \(\mu\). b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Describe all possible alternatives. Which alternative is the best and why?
According to a Gallup poll conducted January \(5-8,2014,67 \%\) of American adults were dissatisfied with the way income and wealth are distributed in America. Assume that this poll is based on a random sample of 1500 American adults. a. What is the point estimate of the corresponding population proportion?
Almost all employees working for financial companies in New York City receive large bonuses at the end of the year. A random sample of 65 employees selected from financial companies in New York City showed that they received an average bonus of \(\$ 55,000\) last year with a standard deviation of \(\$ 18,000 .\) Construct a \(95 \%\) confidence interval for the average bonus that all employees working for financial companies in New York City received last year.
The standard deviation for a population is \(\sigma=7.14\). A random sample selected from this population gave a mean equal to \(48.52\). a. Make a \(95 \%\) confidence interval for \(\mu\) assuming \(n=196\). b. Construct a \(95 \%\) confidence interval for \(\mu\) assuming \(n=100\). c. Determine a \(95 \%\) confidence interval for \(\mu\) assuming \(n=49\). d. Does the width of the confidence intervals constructed in parts a through c increase as the sample size decreases? Explain.
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