6-4 The Central Limit Theorem 287 a. Approach Used for an Individual Value: Use the methods presented in Section 6-2 because we are dealing with an individual value from a normally distributed population. We seek the area of the green-shaded region in Figure 6-23(a). Technology: If using technology (as described at the end of Section 6-2), we find that the green-shaded area is 0.0295. Table A-2: If using Table A-2, we convert the hip width of 16.0 in. to the corresponding z score of z = 1.89, as shown here: z = x - m s = 16.0 - 14.3 0.9 = 1.89 We refer to Table A-2 to find that the cumulative area to the left of z = 1.89 is 0.9706, so the green-shaded area in Figure 6-23 is 1 - 0.9706 = 0.0294. The result of 0.0295 from technology is more accurate. b. Approach Used for the Mean of Sample Values: Because we are dealing with the mean of a sample of 126 males and not an individual male, use the central limit theorem. REQUIREMENT CHECK FOR PART B We can use the normal distribution if the original population is normally distributed or n 7 30. The sample size is greater than 30, so samples of any size will yield means that are normally distributed. Because we are now dealing with a distribution of sample means, we must use the parameters mx and sx, which are evaluated as follows: mx = m = 14.3 sx = s2 n = 0.9 2 126 = 0.1 We want to find the green-shaded area shown in Figure 6-23(b). (Note: Figure 6-23(b) is not drawn to scale. If Figure 6-23(b) had been true to scale, the green-shaded area would not be visible and the normal curve would be much thinner and taller.) Technology: If using technology, the green-shaded area in Figure 6-23(b) is 0+, which is a really small positive number. Table A-2: If using Table A-2, we convert the value of x = 16.0 to the corresponding z score of z = 17.00, as shown here: z = x - mx sx = 16.0 - 14.3 0.1 = 17.00 From Table A-2 we find that the cumulative area to the left of z = 17 is 0.9999, so the green-shaded area of Figure 6-23(b) is 1 - 0.9999 = 0.0001. It is highly unlikely that the 126 adult males will have a mean hip width greater than 16.0 in. YOUR TURN. Do Exercise 5 “Using the Central Limit Theorem.” INTERPRETATION The result from part (a) is more relevant for the design of the aircraft seats. Individual seats will be occupied by individual passengers, not groups of passengers. The result from part (a) shows that with a 0.0295 probability, approximately 3% of adult males would have hip widths greater than the seat width. Although 3% appears to be small, that would result in several passengers on every flight that would somehow require a special accommodation, and that would likely lead to significant challenges for passengers and the flight crew. The reduction of the seat width to 16.0 in. does not appear to be feasible. The Fuzzy Central Limit Theorem In The Cartoon Guide to Statistics, by Gonick and Smith, the authors describe the Fuzzy Central Limit Theorem as follows: “Data that are influenced by many small and unrelated random effects are approximately normally distributed. This explains why the normal is everywhere: stock market fluctuations, student weights, yearly temperature averages, SAT scores: All are the result of many different effects.” People’s heights, for example, are the results of hereditary factors, environmental factors, nutrition, health care, geographic region, and other influences, which, when combined, produce normally distributed values.
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