5.2 EXERCISES SECTION 5.2 Normal Distributions: Finding Probabilities 249 For Extra Help: MyLab Statistics Building Basic Skills and Vocabulary Computing Probabilities for Normal Distributions In Exercises 1–6, the random variable x is normally distributed with mean m = 174 and standard deviation s = 20. Find the indicated probability. 1. P1x 6 1702 2. P1x 6 2002 3. P1x 7 1822 4. P1x 7 1552 5. P1160 6 x 6 1702 6. P1172 6 x 6 1922 Using and Interpreting Concepts Finding Probabilities for Normal Distributions In Exercises 7–12, find the indicated probabilities. If convenient, use technology to find the probabilities. 7. Heights of Women In a survey of 19-year-old U.S. women, their heights were normally distributed, with a mean of 63.9 inches and a standard deviation of 3.3 inches. Find the probability that a randomly selected survey participant has a height that is (a) less than 57 inches, (b) between 60 and 65 inches, and (c) more than 70.5 inches. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Adapted from National Center for Health Statistics) 8. Head Circumference In a survey of male infants from 3 to 5 months of age, head circumference was found to be normally distributed, with a mean of 42.9 centimeters and a standard deviation of 1.5 centimeters. Find the probability that a randomly selected infant from the survey had a head circumference of (a) less than 41 centimeters, (b) between 41 and 45 centimeters, and (c) more than 45.8 centimeters. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Adapted from National Center for Health Statistics) 9. MCAT Scores In a recent year, the MCAT total scores were normally distributed, with a mean of 500.9 and a standard deviation of 10.6. Find the probability that a randomly selected medical student who took the MCAT has a total score that is (a) less than 490, (b) between 490 and 510, and (c) more than 515. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges) 10. MCAT Scores In a recent year, the MCAT scores for the critical analysis and reasoning skills portion of the test were normally distributed, with a mean of 124.8 and a standard deviation of 2.9. Find the probability that a randomly selected medical student who took the MCAT has a critical analysis and reasoning skills score that is (a) less than 120, (b) between 122 and 127, and (c) more than 131. Identify any unusual events in parts (a)–(c). Explain your reasoning. (Source: Association of American Medical Colleges) 11. Utility Bills The monthly utility bills in a city are normally distributed, with a mean of $100 and a standard deviation of $12. Find the probability that a randomly selected utility bill is (a) less than $70, (b) between $90 and $120, and (c) more than $140. 12. Health Club Schedule The amounts of time per workout an athlete uses a stairclimber are normally distributed, with a mean of 20 minutes and a standard deviation of 5 minutes. Find the probability that a randomly selected athlete uses a stairclimber for (a) less than 17 minutes, (b) between 20 and 28 minutes, and (c) more than 30 minutes.
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