91Ó°ÊÓ

A machine at Katz Steel Corporation makes 3 -inch-long nails. The probability distribution of the lengths of these nails is normal with a mean of 3 inches and a standard deviation of \(.1\) inch. The quality control inspector takes a sample of 25 nails once a week and calculates the mean length of these nails. If the mean of this sample is either less than \(2.95\) inches or greater than \(3.05\) inches, the inspector concludes that the machine needs an adjustment. What is the probubility that based on a sample of 25 nails, the inspector will conclude that the machine needs an adjustment?

According to the American Diabetes Association (www.diabetes.org), \(23.1 \%\) of Americans aged 60 years or older had diabetes in 2007 . Assume that this percentage is true for the current population of Americans aged 60 years or older. Let \(\hat{p}\) be the proportion in a random sample of 460 Americans aged 60 years or older who have diabetes. Find the mean and standard deviation of the sampling distribution of \(\hat{p}\), and describe its shape.

Suppose that \(88 \%\) of the cases of car burglar alarms that go off are false. Let \(\hat{p}\) be the proportion of false alarms in a random sample of 80 cases of car burglar alarms that go off. Calculate the mean and standard deviation of \(\hat{p}\), and describe the shape of its sampling distribution.

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 wcight will exceed 6000 pounds? Assume that the 35 people constitute a random sample from the population.