91Ó°ÊÓ

What assumption(s) must hold true to use the normal distribution to make a confidence interval for the population proportion, \(p\) ?

A random sample of 200 observations selected from a population produced a sample proportion equal to \(.91\). a. Make a \(90 \%\) confidence interval for \(p\). b. Construct a \(95 \%\) confidence interval for \(p\). c. Make a \(99 \%\) confidence interval for \(p\). d. Does the width of the confidence intervals constructed in parts a through \(\mathrm{c}\) increase as the confidence level increases? If yes, explain why.

It is said that happy and healthy workers are efficient and productive. A company that manufactures exercising machines wanted to know the percentage of large companies that provide on-site health club facilities. A random sample of 240 such companies showed that 96 of them provide such facilities on site. a. What is the point estimate of the percentage of all such companies that provide such facilities on site? b. Construct a \(97 \%\) confidence interval for the percentage of all such companies that provide such facilities on site. What is the margin of error for this estimate?

In a January 2014 survey conducted by the Associated PressWe TV, \(68 \%\) of American adults said that owning a home is the most important thing or \(a\) very important but not the most important thing (opportunityagenda.org). Assume that this survey was based on a random sample of 900 American adults. a. Construct a \(95 \%\) confidence interval for the proportion of all American adults who will say that owning a home is the most important thing or a very important but not the most important thing. b. Explain why we need to construct a confidence interval. Why can we not simply say that \(68 \%\) of all American adults would say that owning a home is the most important thing or \(a\) very important but not the most important thing?

A consumer agency wants to estimate the proportion of all drivers who wear seat belts while driving. What is the most conservative estimate of the minimum sample size that would limit the margin of error to within \(.03\) of the population proportion for a \(99 \%\) confidence interval?

A random sample of 20 managers was taken, and they were asked whether or not they usually take work home. The responses of these managers are given below, where yes indicates they usually take work home and \(n o\) means they do not. \(\begin{array}{llllllllll}\text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { No } \\ \text { Yes } & \text { Yes } & \text { No } & \text { Yes } & \text { Yes } & \text { No } & \text { No } & \text { No } & \text { No } & \text { Yes }\end{array}\) Make a \(99 \%\) confidence interval for the percentage of all managers who take work home.

A large city with chronic economic problems is considering legalizing casino gambling. The city council wants to estimate the proportion of all adults in the city who favor legalized casino gambling. What is the most conservative estimate of the minimum sample size that would limit the margin of error to be within \(.05\) of the population proportion for a \(95 \%\) confidence interval?

When one is attempting to determine the required sample size for estimating a population mean, and the information on the population standard deviation is not available, it may be feasible to take a small preliminary sample and use the sample standard deviation to estimate the required sample size, \(n .\) Suppose that we want to estimate \(\mu\), the mean commuting distance for students at a community college, to a margin of error within 1 mile with a confidence level of \(95 \%\). A random sample of 20 students yields a standard deviation of \(4.1\) miles. Use this value of the sample standard deviation, \(s\), to estimate the required sample size, \(n .\) Assume that the corresponding population has an approximate normal distribution.