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A population contains 50,000 voters. Use the random number table to identify the voters to be included in a random sample of \(n=15\).

You take a random sample of size \(n=49\) from a distribution with mean \(\mu=53\) and \(\sigma=21\). The sampling distribution of \(\bar{x}\) will be approximately__ with a mean of ___ and a standard deviation (or standard error) of ____

You take a random sample of size \(n=40\) from a distribution with mean \(\mu=100\) and \(\sigma=20 .\) The sampling distribution of \(\bar{x}\) will be approximately _____ with a mean of ____ and a standard deviation (or standard error) of _____

Suppose a random sample of \(n=25\) observations is selected from a population that is normally distributed with mean equal to 106 and standard deviation equal to 12 a. Give the mean and the standard deviation of the sampling distribution of the sample mean \(\bar{x}\). b. Find the probability that \(\bar{x}\) exceeds \(110 .\) c. Find the probability that the sample mean deviates from the population mean \(\mu=106\) by no more than 4

Suppose that college faculty with the rank of professor at two-year institutions earn an average of \(\$ 64,571\) per year \(^{7}\) with a standard deviation of \(\$ 4,000 .\) In an attempt to verify this salary level, a random sample of 60 professors was selected from a personnel database for all two- year institutions in the United States. a. Describe the sampling distribution of the sample \(\operatorname{mean} \bar{x}\) b. Within what limits would you expect the sample average to lie, with probability \(.95 ?\) c. Calculate the probability that the sample mean \(\bar{x}\) is greater than \(\$ 66,000 ?\) d. If your random sample actually produced a sample mean of \(\$ 66,000,\) would you consider this unusual? What conclusion might you draw?

A manufacturer of paper used for packaging requires a minimum strength of 20 pounds per square inch. To check on the quality of the paper, a random sample of 10 pieces of paper is selected each hour from the previous hour's production and a strength measurement is recorded for each. The standard deviation \(\sigma\) of the strength measurements, computed by pooling the sum of squares of deviations of many samples, is known to equal 2 pounds per square inch, and the strength measurements are normally distributed. a. What is the approximate sampling distribution of the sample mean of \(n=10\) test pieces of paper? b. If the mean of the population of strength measurements is 21 pounds per square inch, what is the approximate probability that, for a random sample of \(n=10\) test pieces of paper, \(\bar{x}<20 ?\) c. What value would you select for the mean paper strength \(\mu\) in order that \(P(\bar{x}<20)\) be equal to \(.001 ?\)

The total daily sales, \(x\), in the deli section of a local market is the sum of the sales generated by a fixed number of customers who make purchases on a given day. a. What kind of probability distribution do you expect the total daily sales to have? Explain. b. For this particular market, the average sale per customer in the deli section is \(\$ 8.50\) with \(\sigma=\$ 2.50\). If 30 customers make deli purchases on a given day, give the mean and standard deviation of the probability distribution of the total daily sales, \(x\).

In Exercise 1.67 Allen Shoemaker derived a distribution of human body temperatures with a distinct mound shape. \(^{8}\) Suppose we assume that the temperatures of healthy humans is approximately normal with a mean of 98.6 degrees and a standard deviation of 0.8 degrees. a. If 130 healthy people are selected at random, what is the probability that the average temperature for these people is 98.25 degrees or lower? b. Would you consider an average temperature of 98.25 degrees to be an unlikely occurrence, given that the true average temperature of healthy people is 98.6 degrees? Explain.

Some sports that involve a significant amount of running, jumping, or hopping put participants at risk for Achilles tendinopathy (AT), an inflammation and thickening of the Achilles tendon. A study in The American Journal of Sports Medicine looked at the diameter (in \(\mathrm{mm}\) ) of the affected and nonaffected tendons for patients who participated in these types of sports activities. \(^{9}\) Suppose that the Achilles tendon diameters in the general population have a mean of 5.97 millimeters (mm) with a standard deviation of \(1.95 \mathrm{~mm} .\) a. What is the probability that a randomly selected sample of 31 patients would produce an average diameter of \(6.5 \mathrm{~mm}\) or less for the nonaffected tendon? b. When the diameters of the affected tendon were measured for a sample of 31 patients, the average diameter was \(9.80 .\) If the average tendon diameter in the population of patients with AT is no different than the average diameter of the nonaffected tendons \((5.97 \mathrm{~mm})\), what is the probability of observing an average diameter of 9.80 or higher? c. What conclusions might you draw from the results of part b?

Random samples of size \(n=75\) were selected from a binomial population with \(p=.4 .\) Use the normal distribution to approximate the following probabilities: a. \(P(\hat{p} \leq .43)\) b. \(P(.35 \leq \hat{p} \leq .43)\)