286 CHAPTER 6 Normal Probability Distributions Applying the Central Limit Theorem Many practical problems can be solved with the central limit theorem. Example 2 is a good illustration of the central limit theorem because we can see the difference between working with an individual value in part (a) and working with the mean for a sample in part (b). Study Example 2 carefully to understand the fundamental difference between the procedures used in parts (a) and (b). In particular, note that when working with an individual value, we use z = x - m s , but when working with the mean x for a collection of sample values, we use z = x - m s>2n . Boeing 737 Airline Seats EXAMPLE 2 American Airlines uses Boeing 737 jets with 126 seats in the main cabin. In an attempt to create more room, an engineer is considering a reduction of the seat width from 16.6 in. to 16.0. in. Adult males have hip widths that are normally distributed with a mean of 14.3 in. and a standard deviation of 0.9 in. (based on data from Applied Ergonomics). a. Find the probability that a randomly selected adult male has a hip width greater than the seat width of 16.0 in. b. Find the probability that 126 main cabin seats are all occupied by males with a mean hip width greater than the seat width of 16.0 in. c. For the design of the aircraft seats, which is more relevant: The result from part (a) or the result from part (b)? Why? What do the results suggest about the reduction of the seat width to 16.0 in.? Individual hip widths of adult males m = 14.3 (s = 0.9) x = 16.0 Sample means of hip widths of adult males mx = 14.3 (sx = 0.1) x = 16.0 (a) (b) FIGURE 6-23 Hip Widths of Adult Males SOLUTION
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