Normal Distributions: Finding Probabilities 5.2 246 CHAPTER 5 Normal Probability Distributions What You Should Learn How to find probabilities for normally distributed variables using a table and using technology Probability and Normal Distributions Probability and Normal Distributions When a random variable x is normally distributed, you can find the probability that x will lie in an interval by calculating the area under the normal curve for the interval. To find the area under any normal curve, first convert the upper and lower bounds of the interval to z@scores. Then use the standard normal distribution to find the area. For instance, consider a normal curve with m = 500 and s = 100, as shown at the upper left. The value of x one standard deviation above the mean is m + s = 500 + 100 = 600. Now consider the standard normal curve shown at the lower left. The value of z one standard deviation above the mean is m + s = 0 + 1 = 1. Because a z@score of 1 corresponds to an x@value of 600, and areas are not changed with a transformation to a standard normal curve, the shaded areas in the figures at the left are equal. Finding Probabilities for Normal Distributions A national study found that college students with jobs worked an average of 25 hours per week. The standard deviation is 11 hours. A college student with a job is selected at random. Find the probability that the student works for less than 5 hours per week. Assume that the lengths of time college students work are normally distributed and are represented by the variable x. (Adapted from Statista) SOLUTION The figure shows a normal curve with m = 25, s = 11, and the shaded area for x less than 5. The z@score that corresponds to 5 hours is z = x - m s = 5 - 25 11 ≈ -1.82. The Standard Normal Table shows that P1z 6 -1.822 = 0.0344. The probability that the student works for less than 5 hours per week is 0.0344. Interpretation So, 3.44% of college students with jobs worked for less than 5 hours per week. Because 3.44% is less than 5%, this is an unusual event. TRY IT YOURSELF 1 The average speed of vehicles traveling on a stretch of highway is 67 miles per hour with a standard deviation of 3.5 miles per hour. A vehicle is selected at random. What is the probability that it is violating the speed limit of 70 miles per hour? Assume the speeds are normally distributed and are represented by the variable x. Answer: Page A39 In Example 1, because P1z 6 -1.822 = P1x 6 52, another way to write the probability is P1x 6 52 = 0.0344. EXAMPLE 1 x Hours worked = 25 μ 0 1020304050 200 300 400 500 600 700 800 x 0 −1 −2 −3 1 2 3 z = 0 μ Same area = 500 μ Study Tip To learn how to determine whether a random sample is taken from a normal distribution, see Appendix C.
RkJQdWJsaXNoZXIy NjM5ODQ=