# E there are no outliers because all the observations

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e. There are no outliers because all the observations are within 3 standard deviations of the mean. 34. a. x is 100 and s is 13.88 or approximately 14 b. If the distribution is bell shaped with a mean of 100 points, the percentage of NBA games in which the winning team scores more than 100 points is 50%. A score of 114 points is z = 1 standard deviation above the mean. Thus, the empirical rule suggests that 68% of the winning teams will score between 86 and 114 points. In other words, 32% of the winning teams will score less than 86 points or more than 114 points. Because a bell-shaped distribution is symmetric, approximately 16% of the winning teams will score more than 114 points. c. For the winning margin, x is 11.1 and s is 10.77. To see if there are any outliers, we will first compute the z -score for the winning margin that is farthest from the sample mean of 11.1, a winning margin of 32 points. 32 11.1 1.94 10.77 x x z s = = =

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Chapter 3 3 - 14 Thus, a winning margin of 32 points is not an outlier ( z = 1.94 < 3). Because a winning margin of 32 points is farthest from the mean, none of the other data values can have a z -score that is less than 3 or greater than 3 and hence we conclude that there are no outliers 35. a. x x n i = = = Σ 79 86 20 399 . . Median = 4.17 4.20 4.185 2 + = (average of 10th and 11th values) b. Q 1 = 4.00 (average of 5th and 6th values) Q 3 = 4.50 (average of 15th and 16th values) c. s x x n i = = = Σ ( ) . . 2 1 12 5080 19 08114 d. The distribution is significantly skewed to the left. e. Allison One: z = 412 399 08114 016 . . . . Omni Audio SA 12.3: z = ≈ − 2 32 399 08114 2 06 . . . . f. The lowest rating is for the Bose 501 Series. It’s z -score is: z = ≈ − 214 399 08114 2 28 . . . . This is not an outlier so there are no outliers. 36. 15, 20, 25, 25, 27, 28, 30, 34 Smallest = 15 i = = 25 100 8 2 ( ) Q 1 20 25 2 22 5 = + = . Median = + = 25 27 2 26 i = = 75 100 8 8 ( ) Q 3 28 30 2 29 = + = Largest = 34 37.
Descriptive Statistics: Numerical Methods 3 - 15 15 20 25 30 35 38. 5, 6, 8, 10, 10, 12, 15, 16, 18 Smallest = 5 25 (9) 2.25 100 i = = Q 1 = 8 (3rd position) Median = 10 75 (9) 6.75 100 i = = Q 3 = 15 (7th position) Largest = 18 15 20 5 10 39. IQR = 50 - 42 = 8 Lower Limit: Q 1 - 1.5 IQR = 42 - 12 = 30 Upper Limit: Q 3 + 1.5 IQR = 50 + 12 = 62 65 is an outlier 40. a. Five number summary: 5 9.6 14.5 19.2 52.7 b. IQR = Q 3 - Q 1 = 19.2 - 9.6 = 9.6 Lower Limit: Q 1 - 1.5 (IQR) = 9.6 - 1.5(9.6) = -4.8 Upper Limit: Q 3 + 1.5(IQR) = 19.2 + 1.5(9.6) = 33.6 c. The data value 41.6 is an outlier (larger than the upper limit) and so is the data value 52.7. The financial analyst should first verify that these values are correct. Perhaps a typing error has caused 25.7 to be typed as 52.7 (or 14.6 to be typed as 41.6). If the outliers are correct, the analyst might consider these companies with an unusually large return on equity as good investment candidates. d.

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Chapter 3 3 - 16 20 35 -10 5 50 65 * * 41. a. Median (11th position) 4019 25 (21) 5.25 100 i = = Q 1 (6th position) = 1872 75 (21) 15.75 100 i = = Q 3 (16th position) = 8305 608, 1872, 4019, 8305, 14138 b. Limits: IQR = Q 3 - Q 1 = 8305 - 1872 = 6433 Lower Limit: Q 1 - 1.5 (IQR) = -7777 Upper Limit: Q 3 + 1.5 (IQR) = 17955 c. There are no outliers, all data are within the limits. d. Yes, if the first two digits in Johnson and Johnson's sales were transposed to 41,138, sales would have shown up as an outlier. A review of the data would have enabled the correction of the data.
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• Spring '08
• Staff
• Standard Deviation, Descriptive statistics, standard deviations, x=, Σxi

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