288 CHAPTER 5 Normal Probability Distributions On a dry surface, the braking distances (in feet), from 60 miles per hour to a complete stop, of a sedan can be approximated by a normal distribution, as shown in the figure at the left. Use this information in Exercises 45–50. 45. Find the braking distance that corresponds to z = -2.75. 46. Find the braking distance that corresponds to z = 1.6. 47. What braking distance represents the 90th percentile? 48. What braking distance represents the first quartile? 49. What distance is less than 15% of all the braking distances? 50. What distance is greater than 20% of all the braking distances? Section 5.4 In Exercises 51 and 52, a population and sample size are given. (a) Find the mean and standard deviation of the population. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population. 51. The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2. 52. The minutes of overtime reported by each of the three executives at a corporation are 90, 120, and 210. Use a sample size of 3. In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution. 53. The population densities in people per square mile in the 50 U.S. states have a mean of 199.6 and a standard deviation of 265.4. Random samples of size 35 are drawn from this population, and the mean of each sample is determined. (Source: States101.com) 54. The test scores for the Law School Admission Test (LSAT) in a recent year are normally distributed, with a mean of 151.88 and a standard deviation of 9.95. Random samples of size 40 are drawn from this population, and the mean of each sample is determined. (Source: Law School Admission Council) In Exercises 55– 60, find the indicated probabilities and interpret the results. 55. Refer to Exercise 33. A random sample of 2 years is selected. Find the probability that the mean amount of greenhouse gases for the sample is (a) less than 5500 MMT CO2 eq., (b) between 6000 and 6500 MMT CO2 eq., and (c) greater than 5900 MMT CO2 eq. Compare your answers with those in Exercise 33. 56. Refer to Exercise 34. A random sample of six days is selected. Find the probability that the mean surface concentration of carbonyl sulfide for the sample is (a) between 5.1 and 15.7 picomoles per liter, (b) between 10.5 and 12.3 picomoles per liter, and (c) more than 11.1 picomoles per liter. Compare your answers with those in Exercise 34. x = 132 ft μ = 4.53 ft σ Braking distance (in feet) Braking Distance of a Sedan 115 120 125 130 135 140 145 FIGURE FOR EXERCISES 45–50
RkJQdWJsaXNoZXIy NjM5ODQ=