324 Chapter 7 The Normal Distribution b. A new sample of 80 adults is drawn. Find the probability that more than 40% of the people in this sample have high blood pressure. 0.0256 Tech: 0.0255 c. Find the probability that the proportion of individuals in the sample of 80 who have high blood pressure is between 0.20 and 0.35. 0.8109 Tech: 0.8100 d. Find the probability that less than 25% of the people in the sample of 80 have high blood pressure. 0.1635 Tech: 0.1646 e. Would it be unusual if more than 45% of the individuals in the sample of 80 had high blood pressure? Yes 23. Pay your taxes: According to the Internal Revenue Service, the proportion of federal tax returns for which no tax was paid was p = 0.326. As part of a tax audit, tax officials draw a simple random sample of n = 120 tax returns. a. What is the probability that the sample proportion of tax returns for which no tax was paid is less than 0.30? b. What is the probability that the sample proportion of tax returns for which no tax was paid is between 0.35 and 0.40? 0.2459 Tech: 0.2456 c. What is the probability that the sample proportion of tax returns for which no tax was paid is greater than 0.35? 0.2877 Tech: 0.2874 d. Would it be unusual if the sample proportion of tax returns for which no tax was paid was less than 0.25? Yes 24. Weekly paycheck: The Bureau of Labor Statistics reported that in 2009, the median weekly earnings for people employed full time in the United States was $755. a. What proportion of full-time employees had weekly earnings of more than $755? 0.5 b. A sample of 150 full-time employees is chosen. What is the probability that more than 55% of them earned more than $755 per week? 0.1112 Tech: 0.1103 c. What is the probability that less than 60% of the sample of 150 employees earned more than $755 per week? d. What is the probability that between 45% and 55% of the sample of 150 employees earned more than $755 per week? 0.7776 Tech: 0.7793 e. Would it be unusual if less than 45% of the sample of 150 employees earned more than $755 per week? No 25. Kidney transplants: The Health Resources and Services Administration reported that 5% of people who received kidney transplants were under the age of 18. How large a sample of kidney transplant patients needs to be drawn so that the sample proportion ̂p of those under the age of 18 is approximately normally distributed? 200 26. How’s your new car? The General Social Survey reported that 91% of people who bought a car in the past five years were satisfied with their purchase. How large a sample of car buyers needs to be drawn so that the sample proportion ̂p who are satisfied is approximately normally distributed? 112 Extending the Concepts 27. Flawless tiles: A new process has been designed to make ceramic tiles. The goal is to have no more than 5% of the tiles be nonconforming due to surface defects. A random sample of 1000 tiles is inspected. Let ̂p be the proportion of nonconforming tiles in the sample. a. If 5% of the tiles produced are nonconforming, what is P( ̂p ≥ 0.075)? 0.0001 b. Based on the answer to part (a), if 5% of the tiles are nonconforming, is a proportion of 0.075 nonconforming tiles in a sample of 1000 unusually large? Yes c. If the sample proportion of nonconforming tiles were 0.075, would it be plausible that the goal had been reached? Explain. No d. If 5% of the tiles produced are nonconforming, what is P( ̂p ≥ 0.053)? 0.3300 Tech: 0.3317 e. Based on the answer to part (d), if 5% of the tiles are nonconforming, is a proportion of 0.053 nonconforming tiles in a sample of 1000 unusually large? No f. If the sample proportion of nonconforming tiles were 0.053, would it be plausible that the goal had been reached? Explain. Yes Answers to Check Your Understanding Exercises for Section 7.4 1. ��̂p = 0.82, ��̂p = 0.08591 2. ��̂p = 0.455, ��̂p = 0.03380 3. a. 0.2578 Tech: 0.2580 b. The probability that ̂p is greater than 0.70 is 0.0116 Tech: 0.0115. If we define an event whose probability is less than 0.05 as unusual, then this is unusual. 4. a. 0.8173 Tech: 0.8179 b. The probability that ̂p is less than 0.075 is 0.1190 Tech: 0.1193. This event is not unusual. SECTION 7.5 The Normal Approximation to the Binomial Distribution Objectives 1. Use the normal curve to approximate binomial probabilities Objective 1 Use the normal curve to approximate binomial probabilities We first introduced binomial random variables in Section 6.2. Recall that a binomial random variable represents the number of successes in a series of independent trials. The sample proportion is found by dividing the number of successes by the number of trials.
navidi_monk_elementary_statistics_2e_ch7-9
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