308 CHAPTER 6 Confidence Intervals 50.Ages of College Students An admissions director wants to estimate the mean age of all students enrolled at a college. The estimate must be within 1.5 years of the population mean. Assume the population of ages is normally distributed. (a) Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 1.6 years. (b) The sample mean is 20 years of age. Using the minimum sample size with a 90% level of confidence, does it seem likely that the population mean could be within 7% of the sample mean? within 8% of the sample mean? Explain. 51. Paint Can Volumes A paint manufacturer uses a machine to fill gallon cans with paint (see figure). The manufacturer wants to estimate the mean volume of paint the machine is putting in the cans within 0.5 ounce. Assume the population of volumes is normally distributed. (a) Determine the minimum sample size required to construct a 90% confidence interval for the population mean. Assume the population standard deviation is 0.75 ounce. (b) The sample mean is 127.75 ounces. With a sample size of 8, a 90% level of confidence, and a population standard deviation of 0.75 ounce, does it seem likely that the population mean could be exactly 128 ounces? Explain. 52. Juice Dispensing Machine A beverage company uses a machine to fill half-gallon bottles with fruit juice (see figure). The company wants to estimate the mean volume of water the machine is putting in the bottles within 0.25 fluid ounce. (a) Determine the minimum sample size required to construct a 95% confidence interval for the population mean. Assume the population standard deviation is 1 fluid ounce. (b) The sample mean is exactly 64 fluid ounces. With a sample size of 68, a 95% level of confidence, and a population standard deviation of 1 fluid ounce, does it seem likely that the population mean could be greater than 63.85 fluid ounces? Explain. 53. Soccer Balls A soccer ball manufacturer wants to estimate the mean circumference of soccer balls within 0.15 inch. (a) Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Assume the population standard deviation is 0.5 inch. (b) The sample mean is 27.5 inches. With a sample size of 84, a 99% level of confidence, and a population standard deviation of 0.5 inch, does it seem likely that the population mean could be less than 27.6 inches? Explain. 54. Tennis Balls A tennis ball manufacturer wants to estimate the mean circumference of tennis balls within 0.05 inch. Assume the population of circumferences is normally distributed. (a) Determine the minimum sample size required to construct a 99% confidence interval for the population mean. Assume the population standard deviation is 0.10 inch. (b) The sample mean is 8.3 inches. With a sample size of 34, a 99% level of confidence, and a population standard deviation of 0.10 inch, does it seem likely that the population mean could be exactly 8.258 inches? Explain. Volume = 1 gal (128 oz) Error tolerance = 0.5 oz FIGURE FOR EXERCISE 51 Orange Juice with Pulp 64 fl oz Volume = 1/2 gal (64 fl oz) Error tolerance = 0.25 fl oz FIGURE FOR EXERCISE 52
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