91Ó°ÊÓ

What is the estimator of the population proportion? Is this estimator an unbiased estimator of \(p ?\) Explain why or why not.

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 percentage is true for the current population of American adults. Let \(\hat{p}\) be the proportion in a random sample of 1000 American adults who hold the above opinion. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\) and describe its shape.

According to the American Time Use Survey results released by the Bureau of Labor Statistics on June 24,2015, on a typical day, \(65 \%\) of American men age 15 and over spent some time doing household activities such as housework, cooking, lawn care, or financial and other household management. Assume that this percentage is true for the current population of all American men age 15 and over. A random sample of 600 American men age 15 and over is selected. a. Find the probability that the sample proportion is \(\begin{array}{ll}\text { i. less than .68 } & \text { ii. between } .63 \text { and } .69\end{array}\) b. What is the probability that the sample proportion is within \(.025\) of the population proportion? c. What is the probability that the sample proportion is greater than the population proportion by \(.03\) or more?

The test scores for 300 students were entered into a computer, analyzed, and stored in a file. Unfortunately, someone accidentally erased a major portion of this file from the computer. The only information that is available is that \(30 \%\) of the scores were below 65 and \(15 \%\) of the scores were above 90 . Assuming the scores are approximately normally distributed, find their mean and standard deviation.

A television reporter is covering the election for mayor of a large city and will conduct an exit poll (interviews with voters immediately after they vote) to make an early prediction of the outcome. Assume that the eventual winner of the election will get \(60 \%\) of the votes. a. What is the probability that a prediction based on an exit poll of a random sample of 25 voters will be correct? In other words, what is the probability that 13 or more of the 25 voters in the sample will have voted for the eventual winner? b. How large a sample would the reporter have to take so that the probability of correctly predicting the outcome would be \(.95\) or higher?

A certain elevator has a maximum legal carrying capacity of 6000 pounds. Suppose that the population of all people who ride this elevator have a mean weight of 160 pounds with a standard deviation of 25 pounds. If 35 of these people board the elevator, what is the probability that their combined weight will exceed 6000 pounds? Assume that the 35 people constitute a random sample from the population.