316 Chapter 7 The Normal Distribution c. If appropriate find the 80th percentile of ̄x. 64.51 Tech: 64.52 18. A sample of size 15 will be drawn from a population with mean 125 and standard deviation 28. a. Is it appropriate to use the normal distribution to find probabilities for ̄x? No b. If appropriate find the probability that ̄x will be less than 120. Not appropriate c. If appropriate find the 90th percentile of ̄x. Not appropriate Working with the Concepts 19. Summer temperatures: Following are the temperatures, in degrees Fahrenheit, in Denver for five days in July 2009: Date Temperature July 21 69 July 22 75 July 23 79 July 24 83 July 25 71 a. Consider this to be a population. Find the population mean �� and the population standard deviation ��. b. List all samples of size 2 drawn with replacement. There are 5 × 5 = 25 different samples. c. Compute the sample mean ̄x for each of the 25 samples of size 2. Compute the mean ��̄ x and the standard deviation ��̄ x of the sample means. d. Verify that ��̄ x = �� and ��̄ x = ��∕ √ 2. 20. Ages of winners: Following are the ages of the Grammy Award winners for Best New Artist for the years 2009–2013. (For the Zac Brown Band, the age given is that of lead singer Zac Brown. For Bon Iver, the age is that of lead singer Justin Vernon. For Fun., the age is that of guitarist Andrew Dost.) Year: Winner Age 2013: Fun. 29 2012: Bon Iver 30 2011: Esperanza Spalding 26 2010: Zac Brown Band 31 2009: Adele 20 a. Consider this to be a population. Find the population mean �� and the population standard deviation ��. b. List all samples of size 2 drawn with replacement. There are 5 × 5 = 25 different samples. c. Compute the sample mean ̄x for each of the 25 samples of size 2. Compute the mean ��̄ x and the standard deviation ��̄ x of the sample means. d. Verify that ��̄ x = �� and ��̄ x = ��∕ √ 2. 21. How’s your mileage? The Environmental Protection Agency (EPA) rates the mean highway gas mileage of the 2013 Ford Edge to be 27 miles per gallon. Assume the standard deviation is 3 miles per gallon. A rental car company buys 60 of these cars. a. What is the probability that the average mileage of the fleet is greater than 26.5 miles per gallon? 0.9015 Tech: 0.9016 b. What is the probability that the average mileage of the fleet is between 26 and 26.8 miles per gallon? 0.2966 Tech: 0.2979 c. Would it be unusual if the average mileage of the fleet were less than 26 miles per gallon? Yes 22. Watch your cholesterol: The National Health and Nutrition Examination Survey (NHANES) reported that in a recent year, the mean serum cholesterol level for U.S. adults was 202, with a standard deviation of 41 (the units are milligrams per deciliter). A simple random sample of 110 adults is chosen. a. What is the probability that the sample mean cholesterol level is greater than 210? 0.0202 Tech: 0.0204 b. What is the probability that the sample mean cholesterol level is between than 190 and 200? 0.3039 Tech: 0.3034 c. Would it be unusual for the sample mean to be less than 198? No 23. TV sets: According to the Nielsen Company, the mean number of TV sets in a U.S. household in 2013 was 2.24. Assume the standard deviation is 1.2. A sample of 85 households is drawn. a. What is the probability that the sample mean number of TV sets is greater than 2? 0.9671 Tech: 0.9674 b. What is the probability that the sample mean number of TV sets is between 2.5 and 3? 0.0228 Tech: 0.0229 c. Find the 30th percentile of the sample mean. 2.17 d. Would it be unusual for the sample mean to be less than 2? Yes e. Can you tell whether it would be unusual for an individual household to have fewer than 2 TV sets? Explain. No 24. SAT scores: The College Board reports that in 2013, the mean mathematics SAT score was 514, and the standard deviation was 118. A sample of 65 scores is chosen. a. What is the probability that the sample mean score is less than 500? 0.1685 Tech: 0.1694 b. What is the probability that the sample mean score is between 480 and 520? 0.6489 Tech: 0.6490 c. Find the 80th percentile of the sample mean. 526.3 d. Would it be unusual if the sample mean were greater than 550? Yes e. Can you tell whether it would be unusual for an individual to get a score greater than 550? Explain. No 25. Taxes: The Internal Revenue Service reports that the mean federal income tax paid in the year 2010 was $8040. Assume that the standard deviation is $5000. The IRS plans to draw a sample of 1000 tax returns to study the effect of a new tax law. a. What is the probability that the sample mean tax is less than $8000? 0.4013 Tech: 0.4001 b. What is the probability that the sample mean tax is between $7600 and $7900? 0.1840 Tech: 0.1853 c. Find the 40th percentile of the sample mean. 8000.5 Tech: 7999.9 d. Would it be unusual if the sample mean were less than $7500? Yes e. Can you tell whether it would be unusual for an individual to pay a tax of less than $7500? Explain. No 26. High-rent district: The Real Estate Group NY reports that the mean monthly rent for a one-bedroom apartment without a doorman in Manhattan is $2631. Assume the standard deviation is $500. A real estate firm samples 100 apartments. a. What is the probability that the sample mean rent is greater than $2700? 0.0838 b. What is the probability that the sample mean rent is between $2500 and $2600? 0.2632
navidi_monk_elementary_statistics_2e_ch7-9
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