272 CHAPTER 5 Normal Probability Distributions Finding Probabilities for Sampling Distributions In Exercises 29–32, find the indicated probability and interpret the results. 29. Dow Jones Industrial Average From 1975 through 2020, the mean annual gain of the Dow Jones Industrial Average was 652. A random sample of 32 years is selected from this population. What is the probability that the mean gain for the sample was between 400 and 700? Assume s = 1540. 30. Standard & Poor’s 500 From 1921 through 2020, the mean return of the Standard & Poor’s 500 was 12.59%. A random sample of 38 years is selected from this population. What is the probability that the mean return for the sample was between 10.0% and 11.0%? Assume s = 19.58%. 31. Asthma Prevalence by State The mean percent of asthma prevalence of the 50 U.S. states is 9.51%. A random sample of 30 states is selected. What is the probability that the mean percent of asthma prevalence for the sample is greater than 10%? Assume s = 1.17%. (Source: Centers for Disease Control and Prevention) 32. Fertility Rates by State A fertility rate is the number of births per 1000 women aged 15–44. The mean fertility rate of the 50 U.S. states for a recent year was 59.0 with a standard deviation of 5.47. A random sample of 40 states is selected. What is the probability that the mean fertility rate for the sample for the recent year was less than 58? (Source: Centers for Disease Control and Prevention) 33. Which Is More Likely? Assume that the asthma prevalences in Exercise 31 are normally distributed. Are you more likely to randomly select a state with asthma prevalence less than 10% or to randomly select a sample of 10 states in which the mean of the state asthma prevalences is less than 10%? Explain. 34. Which Is More Likely? Assume that the fertility rates in Exercise 32 are normally distributed. Are you more likely to randomly select a state with a fertility rate of less than 65 or to randomly select a sample of 15 states in which the mean of the state fertility rates is less than 65? Explain. 35. Paint Cans A machine is set to fill paint cans with a mean of 128 ounces and a standard deviation of 0.2 ounce. A random sample of 40 cans has a mean of 127.9 ounces. The machine needs to be reset when the mean of a random sample is unusual. Does the machine need to be reset? Explain. 36. Milk Containers A machine is set to fill milk containers with a mean of 64 ounces and a standard deviation of 0.11 ounce. A random sample of 40 containers has a mean of 64.05 ounces. The machine needs to be reset when the mean of a random sample is unusual. Does the machine need to be reset? Explain. 37. Lumber Cutter The lengths of lumber a machine cuts are normally distributed, with a mean of 96 inches and a standard deviation of 0.5 inch. (a) What is the probability that a randomly selected board cut by the machine has a length greater than 96.25 inches? (b) You randomly select 40 boards. What is the probability that their mean length is greater than 96.25 inches? 38. Ice Cream The weights of ice cream cartons are normally distributed with a mean weight of 10 ounces and a standard deviation of 0.5 ounce. (a) What is the probability that a randomly selected carton has a weight greater than 10.21 ounces? (b) You randomly select 25 cartons. What is the probability that their mean weight is greater than 10.21 ounces?
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