12.5 The Normal Curve 821 88. Consider the following two normal curves. 28 30 32 34 36 38 40 42 44 A B a) Do these distributions have the same mean? If so, what is the mean? Yes, 36 b) One of these curves corresponds to a normal distribution with a standard deviation of 1. The other curve corresponds to a normal distribution with a standard deviation of 3. Which curve, A or B, has a standard deviation of 3? B; since curve B is more spread out it has the higher standard deviation. Concept/Writing Exercises 89. In a distribution that is skewed to the right, which has the greatest value: the mean, median, or mode? Which has the smallest value? Explain. The mean is the greatest value. The median is lower than the mean. The mode is the lowest value. 90. In a distribution skewed to the left, which has the greatest value: the mean, median, or mode? Which has the smallest value? Explain. The mode is the highest value. The median is lower than the mode. The mean is the lowest value. 91. List three populations other than those given in the text that may be normally distributed. Answers will vary. 92. List three populations other than those given in the text that may not be normally distributed. Answers will vary. Challenge Problems/Group Activities 93. Salesperson Promotion The owner at Kim’s Home Interiors is reviewing the sales records of two managers who are up for promotion, Katie and Stella, who work in different stores. At Katie’s store, the mean sales have been $23,200 per month, with a standard deviation of $2170. At Stella’s store, the mean sales have been $25,600 per month, with a standard deviation of $2300. Last month, Katie’s store sales were $28,408 and Stella’s store sales were $29,510. At both stores, the distribution of monthly sales is normal. a) Convert last month’s sales for Katie’s store and for Stella’s store to z-scores. Katie: = z 2.4; Stella: = z 1.7 b) If one of the two were to be promoted based solely on the sales last month, who should be promoted? Explain. Katie. Her z-score is higher than Stella’s z-score, which means her sales are further above the mean than Stella’s sales. 94. Chebyshev’s Theorem How can you determine whether a distribution is approximately normal? A statistical theorem called Chebyshev’s theorem states that the minimum percent of data between plus and minus K standard deviations from the mean > K( 1) in any distribution can be determined by the formula = − K Minimum percent 1 1 2 Thus, for example, between ±2 standard deviations from the mean there will always be a minimum of 75% of data. This minimum percent applies to any distribution. For = K 2, = − = − = Minimum percent 1 1 2 1 1 4 3 4 , or 75% 2 Likewise, between ±3 standard deviations from the mean there will always be a minimum of 89% of the data. For = K 3, = − = − = Minimum percent 1 1 3 1 1 9 8 9 , or 89% 2 The following table lists the minimum percent of data in any distribution and the actual percent of data in the normal distribution between ± ± ± 1.1, 1.5, 2.0, and ±2.5 standard deviations from the mean. The minimum percents of data in any distribution were calculated by using Chebyshev’s theorem. The actual percents of data for the normal distribution were calculated by using the area given in the standard normal, or z, table. = K 11. = K 15. = K 2 = K 25. Minimum (for any distribution) 17.4% 55.6% 75% 84% Normal distribution 72.9% 86.6% 95.4% 98.8% Given distribution The percentages for the third row of the chart will be calculated in part (d). Consider the following 30 pieces of data obtained from a quiz. 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10 a) Determine the mean of the set of scores. ≈ 5.33 b) Determine the standard deviation of the set of scores. ≈ 8.99 3.00 c) Determine the values that correspond to 1.1, 1.5, 2.0, and 2.5 standard deviations above the mean. Then $ Digital Vision/Photodisc/ Getty Images
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