8.7 A random variable is normally distributed with mean $1500 and standard deviation $100. Determine the standard error of the sampling distribution of the mean for simple random samples with the following sample sizes: a. n=5 16 b. n=100 c. n=400 d. n=1000 8.9 For a random variable that is normally distributed, with u= 80 and Q= 10, determine the probability that a simple random sample of 25 items will have a mean that is a. greater than 78. b. between 79 and 85. c. less than 85. 8.14 From past experience, an airline has found the luggage weight for individual air travelers on its trans- Atlantic route to have a mean of 80 pounds and a standard deviation of 20 pounds. The plane is consistently fully booked and holds 100 passengers. The pilot insists on loading an extra 500 pounds of fuel whenever the total luggage weight exceeds 8300 pounds. On what percentage of the flights will she end up having the extra fuel loaded? 8.25 It has been reported that 40% of U.S. workers employed as purchasing managers are females. In a simple random sample of U.S. purchasing managers, 70 out of the 200 are females. Given this information: Source: Bureau of the Census, Statistical Abstract of the United States 2009, p. 384. a. What is the population proportion, pie? b. What is the sample proportion, p? c. What is the standard error of the sample proportion? d. In the sampling distribution of the proportion, what is the probability that a sample of this size would result in a sample proportion at least as large as that found in part (b)? 8.46 Based on past experience, a telemarketing firm has found that calls to prospective customers take an average of 2.0 minutes, with a standard deviation of 1.5 minutes. The distribution is positively skewed, since persons who actually become customers require more of the caller's time than those who are not home, who simply hang up, or who say they're not interested. Albert has been given a quota of 220 calls for tomorrow, and he works an 8-hour day. Assuming that his list represents a simple random sample of those persons who could be called, what is the probability that Albert will meet or exceed his quota? 8.47 When a production machine is properly calibrated, it requires an average of 25 seconds per unit produced, with a standard deviation of 3 seconds. For a simple random sample of n =36 units, the sample mean is found to be}x 5 26.2 seconds per unit. a. What z-score corresponds to the sample mean of }x=5 26.2 seconds? b. When the machine is properly calibrated, what is the probability that the mean for a simple random sample of this size will be at least 26.2 seconds? c. Based on the probability determined in part (b), does it seem likely that the machine is properly calibrated? Explain your reasoning 8.51 Based on past experience, 20% of the contacts made by a firm's sales representatives result in a sale being made. Charlie has contacted 100 potential customers but has made only 10 sales. Assume that Charlie's contacts represent a simple random sample of those who could have been called upon. Given this information: a. What is the sample proportion, p 5 proportion of contacts that resulted in a sale being made? b. For simple random samples of this size, what is the probability that p <0.10? c. Based on your answer to (b), would you tend to accept Charlie's explanation that he \"just had a bad day\"? 8.52 As a project for their high school mathematics class, four class members purchase packages of a leading breakfast cereal for subsequent weighing and comparison with the 12-ounce label weight. The students are surprised when they weigh the contents of their four packages and find the average weight to be only 11.5 ounces. When they write to the company, its consumer relations representative explains that \"the contents of the '12-ounce' packages average 12.2 ounces and have a standard deviation of 0.4 ounces. The amount of cereal will vary from one box to the next, and your sample is very smallthis explains why your average weight could be just 11.5 ounces.\" If the spokesperson's information is correct, how unusual would it be for these students to get a sample mean as small as the one they obtained? Be sure to state any assumptions you have made in obtaining your answer. 8.58 The overall pass rate for law school graduates taking the Maryland bar exam has been reported as 76%. Assume that a certain Maryland law school has had 400 of its most recent graduates take the Maryland bar exam, but only 60% passed. When asked about these results, the dean of this university's law school claims his graduates are just as good as others who have taken the Maryland bar exam, and the low pass rate for the recent graduates was just \"one of those statistical fluctuations that happen all the time.\" If this university's overall pass rate for the Maryland bar exam were really 76%, what would be the probability of a simple random sample of 400 having a pass rate of 60% or less? Based on this probability, comment on the dean's explanation