Section 7.3 Sampling Distributions and the Central Limit Theorem 317 c. Find the 60th percentile of the sample mean. d. Can you tell whether it would be unusual if the sample mean were greater than $2800? Yes e. Do you think it would be unusual for an individual apartment to have a rent greater than $2800? Explain. No 27. Roller coaster ride: A roller coaster is being designed that will accommodate 60 riders. The maximum weight the coaster can hold safely is 12,000 pounds. According to the National Health Statistics Reports, the weights of adult U.S. men have mean 194 pounds and standard deviation 68 pounds, and the weights of adult U.S. women have mean 164 pounds and standard deviation 77 pounds. a. If 60 people are riding the coaster, and their total weight is 12,000 pounds, what is their average weight? 200 pounds b. If a random sample of 60 adult men ride the coaster, what is the probability that the maximum safe weight will be exceeded? 0.2483 Tech: 0.2472 c. If a random sample of 60 adult women ride the coaster, what is the probability that the maximum safe weight will be exceeded? 0.0001 28. Elevator ride: Engineers are designing a large elevator that will accommodate 40 people. The maximum weight the elevator can hold safely is 8120 pounds. According to the National Health Statistics Reports, the weights of adult U.S. men have mean 194 pounds and standard deviation 68 pounds, and the weights of adult U.S. women have mean 164 pounds and standard deviation 77 pounds. a. If 40 people are on the elevator, and their total weight is 8120 pounds, what is their average weight? 203 pounds b. If a random sample of 40 adult men ride the elevator, what is the probability that the maximum safe weight will be exceeded? 0.2005 Tech: 0.2013 c. If a random sample of 40 adult women ride the elevator, what is the probability that the maximum safe weight will be exceeded? 0.0007 29. Annual income: The mean annual income for people in a certain city (in thousands of dollars) is 42, with a standard deviation of 30. A pollster draws a sample of 90 people to interview. a. What is the probability that the sample mean income is less than 38? 0.1038 Tech: 0.1030 b. What is the probability that the sample mean score is between 40 and 45? 0.5646 Tech: 0.5651 c. Find the 60th percentile of the sample mean. d. Would it be unusual for the sample mean to be less than 35? Yes e. Can you tell whether it would be unusual for an individual to have an income less than 35? Explain. No 30. Going to work: An ABC News report stated that the mean distance that commuters in the United States travel each way to work is 16 miles. Assume the standard deviation is 8 miles. A sample of 75 commuters is chosen. a. What is the probability that the sample mean commute distance is greater than 13 miles? 0.9994 b. What is the probability that the sample mean commute distance is between 18 and 20 miles? 0.0150 Tech: 0.0152 c. Find the 10th percentile of the sample mean. 14.82 d. Would it be unusual for the sample mean distance to be greater than 19 miles? Yes e. Can you tell whether it would be unusual for an individual to have a commute distance greater than 19 miles? Explain. No Extending the Concepts 31. Eat your cereal: A cereal manufacturer claims that the weight of a box of cereal labeled as weighing 12 ounces has a mean of 12.0 ounces and a standard deviation of 0.1 ounce. You sample 75 boxes and weigh them. Let ̄x denote the mean weight of the 75 boxes. a. If the claim is true, what is P(̄x ≤ 11.99)? b. Based on the answer to part (a), if the claim is true, is 11.99 ounces an unusually small mean weight for a sample of 75 boxes? No c. If the mean weight of the boxes were 11.99 ounces, would you be convinced that the claim was false? Explain. No d. If the claim is true, what is P(̄x ≤ 11.97)? 0.0047 e. Based on the answer to part (d), if the claim is true, is 11.97 ounces an unusually small mean weight for a sample of 75 boxes? Yes f. If the mean weight of the boxes were 11.97 ounces, would you be convinced that the claim was false? Explain. Yes 32. Battery life: A battery manufacturer claims that the lifetime of a certain type of battery has a population mean of �� = 40 hours and a standard deviation of �� = 5 hours. Let ̄x represent the mean lifetime of the batteries in a simple random sample of size 100. a. If the claim is true, what is P(̄x ≤ 38.5)? 0.0013 b. Based on the answer to part (a), if the claim is true, is a sample mean lifetime of 38.5 hours unusually short? Yes c. If the sample mean lifetime of the 100 batteries were 38.5 hours, would you find the manufacturer’s claim to be plausible? Explain. No d. If the claim is true, what is P(̄x ≤ 39.8)? 0.3446 e. Based on the answer to part (d), if the claim is true, is a sample mean lifetime of 39.8 hours unusually short? No f. If the sample mean lifetime of the 100 batteries were 39.8 hours, would you find the manufacturer’s claim to be plausible? Explain. Yes 33. Finite population correction: The mean of a sample of size n has standard deviation ��∕ √ n, where �� is the population standard deviation. When sampling without replacement, a more accurate expression can be obtained by multiplying by a correction factor. Specifically, if the sample size is more than 5% of the population size, it is better to compute the standard deviation of the sample mean as �� √ n √ N − n N − 1 where √ N is the population size and n is the sample size. The factor N − n N − 1 is called the finite population correction factor. a. One hundred students took an exam. The standard deviation of the 100 scores was 10. Twenty exams were
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