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CHAPTER 3 DESCRIBING DATA: NUMERICAL MEASURES 1. 5.4 found by 27/5 2. 55 . found by 33/6 3. a. b. Mean = 7.0, found by 28/4 (5 7) (9 7) (4 7) (10 7) 0 4. a. b. 4.2 found by 21/5 (13 . 4.2) (7.0 4.2) (3.6 4.2) (4.1 4.2) (5.0 4.2) 0 5. 14.58, found by 43.74/3 6. $20.95, found by $125.68/6 7. a. b. 15.4, found by 154/10 Population parameter since it includes all the salespersons at Midtown Ford. 8. a. b. 23.9, found by 167/7 Population parameter since it includes all the calls during a seven-day period. 9. a. b. $54.55, found by $1091/20 A sample statistic, assuming that the power company serves more than 20 customers. 10. a. b. 10.73, found by 161/15 Sample of RN’s 11. $22.91, found by 12. $1.50 found by ($40 + $35)/50 13. $11.50, found by ($400 + $500 + $1400)/200 14. $143.75, found by ($1000 +$750 + $4000)/40 15. a. b. c. 16. Median = 33, Mode = 15 17. Median = 5 Mode = 5 18. Median = 10.5 Mode = 8 Chapter 3 300($20) 400($25) 400($23) 300 400 400 no mode The given value would be the mode 3 and 4, bimodal 24 $25, 200 1100 19. a. b. Median = 71.75% Mode = 69.4% 20. Median = 9.2 Modes are 8.2, 8.5, and 10.3 21. 12.8% increase found by 22. 6% increase found by 23. 12.28% increase found by 5 (1.094)(1.138)(1.117)(1.119)(1.147) 1 24. 54.48% increase found by 3 2, 400,000,000 1 651,000,000 25. 10.33% increase found by 26. GM = 10 27. 10.76% found by 28. 5.49% for public colleges found by 29. a. b. c. d. 7, found by 10 – 3 6, found by 30/5 2.4, found by 12/5 The difference between the highest number sold (10) and the smallest number sold (3) is 7. On the average the number of service reps on duty deviates by 2.4 from the mean of 6. 30. a. b. c. d. 24, found by 52 – 28 38 6.25, found by 50/8 The difference between 28 and 52 is 24. On the average the number of students enrolled deviates 6.25 from the mean of 38. 31. a. b. c. d. 30, found by 54 – 24 38, found by 380/10 7.2, found by 72/10 The difference between 54 and 24 is 30. On the average the number of minutes required to install a door deviates 7.2 minutes from the mean of 38 minutes. 8 5 (1.08)(1.12)(1.14)(1.26)(1.05) 1 (1.02)(1.08)(1.06)(1.04)(1.10)(1.06)(1.08)(1.04) 1 13 14.0 1 3.9 54.87 1 0.1956 9.19 5 70 1 42 8954 1 4975 22,608 1 5.70% for private colleges found by 11 12, 284 The rates of increase are about the same with private colleges slightly higher. 11 25 Chapter 3 32. a. b. c. d. 7.6%, found by 18.2 – 10.6 13.85% found by 110.8/8 2%, found by 16/8 The difference between 18.2 and 10.6 is 7.6%. On the average the return on investment deviates two percent from the mean of 13.85%. 33. a. b. c. d. 15, found by 41 – 26 33.9, found by 339/10 4.12, found by 41.2/10 The ratings deviate 4.12 from the mean of 33.9 on the average. 34. a. b. c. d. 10 days, found by 10 – 0 3.5 days found by 28/8 2.375 days, found by 19/8 Days lost to illness deviates 2.375 days on average from the mean. The difference of 10 and 0 is 10. 35. a. 5 b. 4.4 found by a. 8 b. 9.67 found by a. $2.77 b. 2 38. a. b. 11.76%, found by 58.8/5 16.89, found by 84.452/5 39. a. Range = 7.3, found by 11.6 – 4.3 Arithmetic mean = 6.94, found by 34.7/5 Variance = 6.5944, found by 32.972/5 Standard Deviation = 2.568 Dennis has a higher mean return (11.76 > 6.94). However, Dennis has greater spread in their returns on equity (16.89 > 6.59). 36. 37. b. (8 5) 2 (3 5) 2 (7 5) 2 (3 5) 2 ( 4 5) 2 5 (13 8) 2 (3 8) 2 (8 8) 2 (10 8) 2 (8 8) 2 (6 8) 2 6 (2.68 2.77) 2 ... (4.30 2.77) 2 (358 . 2.77) 2 126 . 5 40. a. b. c. d. $18,000, found by $90,000 – 72,000 $79,600, found by $398,000/5 Variance = 40,240,000, found by 201,200,000/5 Standard Deviation = $6343.50 Means about the same, but less dispersion in salary for TMV vice presidents. 41. a. X 4 s2 (20) 2 5 5.50 5 1 102 b. s2 c. s = 2.3452 Chapter 3 (7 4)2 ...(3 4)2 5.5 5 1 26 42. 43. 44. 45. a. X 8 s2 (11 8)2 ...(7 8)2 5.5 5 1 (40)2 5 5.50 5 1 342 b. s2 c. s = 2.3452 a. X 38 s2 (28 38)2 ...(42 38)2 82.6667 10 1 (380) 2 10 82.6667 10 1 15,184 b. s2 c. s = 9.0921 a. X 13.85 s2 (10.6 13.85)2 ...(15.6 13.85)2 6.0086 8 1 (110.8) 2 8 6.0086 8 1 1576.64 b. s2 c. s = 2.4512 a. X 124 s2 (124 124)2 ...(121 124)2 4.6667 10 1 (1240) 2 10 4.6667 10 1 153,802 b. s2 c. s = 2.1602 46. AB4 yields a higher mean weight with less spread. 47. a. b. About 95% 47.5%, 2.5% 48. a. b. c. d. 85, halfway between the endpoints of 140 and 30 About 18, found by (140 – 30)/6 103 and 67, found by 85 (1)18 121 and 49, found by 85 (2)18 49. 8.06%, found by (0.25/3.10)(100) 50. Domestic 23.81%, found by (5/21)(100). Overseas 20%, found by (7/35)(100). There is slightly more relative dispersion in the weights of luggage for domestic passengers. 51. a. b. Because the two series are in different units of measurement. P.E. ratio = 16.51% ROI 20.8%, less spread in the P.E. ratios 52. a. b. The data are in the same units but the means, relatively speaking, are far apart. The relative dispersion in stocks under $10 is 28.95%. For stocks over $60, 5.71%. Less relative dispersion in stocks over $60. 27 Chapter 3 53. a. The mean is 30.8, found by 154/5. The median is 31.0 and the standard deviation is 3.96, 4806 found as b. 54. a. 154 2 5 . 4 3(30.8 31.0) 0.15, found by . 3.96 The mean is 542, found by 8130/15. The median is 546 and the standard deviation is 4,415,268 25.08, found as b. 55. a. b. 56. a. 14 3(542 546) –0.478, found by 25.08 The mean is 21.93, found by 328.9 / 15. The median is 15.8 and the standard deviation is 328.9 2 13,494.676 15 . 21.18, found as 14 3(21.93 15.8) 0.868, found by . 21.18 The mean is 3.046, found by 88.34/29. The median is 2.00 and the standard deviation is 514.9526 2.963, found as b. 8130 2 15 . 28 3(3.046 2) 1.059, found by 2.963 88.34 2 29 . 57. Median = 53 found by (11 + 1)(1/2) therefore 6th value in from lowest. Q1 = 49 found by (11 + 1)(1/4) therefore 3rd value in from lowest Q3 = 55 found by (11 + 1)(3/4) therefore 9th value in from lowest 58. Median = 9.53, found by (9.45 + 9.61)/2 Q1 = 7.69 found by 7.59 + (7.99 – 7.59) ¼ Q3 = 12.59 found by 12.22 + (12.71 – 12.22)3/4 59. a. b. c. Q1 = 33.25 D2 = 27.8 P67 = 47 60. a. b. c. d. Median = 58 Q1 = 51.25 D1 = 45.3 P33 = 53.53 Chapter 3 Q3 = 50.25 D8 = 52.6 Q3 = 66.0 D9 = 76.4 28 61. a. b. c. d. e. f. 350 Q1 = 175 Q3 = 930 755, found by 930 – 175 Less than zero, or more than about 2060 There are no outliers The distribution is positively skewed 62. a. b. c. d. e. f. 450 Q1 = 300 Q3 = 700 400, found by 700 – 300 Less than zero or more than 1300 One outlier at about 1500 Distribution is positively skewed 63. The distribution is somewhat positively skewed. Note that dashed line above 15.5 is longer than below 7.8. BoxPlot 0 5 10 15 20 25 64. The median is $253, the smallest value is $116 and the largest is $353. About 25% of the semiprivate rooms are less than $214 and 25% above $304. The distribution is negatively skewed. 65. Because the exact values in a frequency distribution are not known, the midpoint of the class is used for every member of that class. 29 Chapter 3 66. Class 0 up to 5 5 up to 10 10 up to 15 15 up to 20 20 up to 25 Total X 67. 68. 70. X 25 35 45 55 65 s M 15 25 35 45 55 2410 40.17 60 s f 1 15 22 8 4 50 2240 44.8 50 Expenditure 25 up to 35 35 up to 45 45 up to 55 55 up to 65 Chapter 3 f 7 12 21 18 12 70 f 3 7 18 20 12 60 Amount 20 up to 30 30 up to 40 40 up to 50 50 up to 60 60 up to 70 Total X M 2.5 7.5 12.5 17.5 22.5 s 3310 47.2857 70 Age 10 up to 20 20 up to 30 30 up to 40 40 up to 50 50 up to 60 Total X 69. 380 12.67 30 Class 20 up to 30 30 up to 40 40 up to 50 50 up to 60 60 up to 70 Total X f 2 7 12 6 3 30 X 25 35 45 55 65 s f 5 10 21 16 X 30 40 50 60 fM 5.00 52.50 150.00 105.00 67.50 380.00 fM2 12.50 393.75 1875.00 1837.50 1518.75 5637.50 5637.5 (380)2 / 30 5.33 29 M 175 420 945 990 780 3310 fM2 4375 14,700 42,525 54,450 50,700 166,750 166,750 (3310)2 / 70 12.179 69 fM 45 175 630 900 660 2410 fM2 675 4375 22,050 40,500 36,300 103,900 103,900 (2410)2 / 60 10.97 60 1 M 25 525 990 440 260 2240 fM2 625 18,375 44,550 24,200 16,900 104,650 104,650 (2240) 2 / 50 9.37 50 1 M fM2 150 4500 400 16,000 1050 52,500 960 57,600 30 65 up to 75 Total X 71. 8 60 3120 52 60 s 560 39,200 3120 169,800 169,800 (3120)2 / 60 11.32 60 1 b. c. Mean = 5, found by (6 + 4 + 3 + 7 + 5)/5 Median is 5, found by rearranging the values and selecting the middle value. Population because all partners were included. ( X ) (6 5) (4 5) (3 5) (7 5) (5 5) 0 72. a. b. Mean = 21.71, Median = 22.00 (23 21.7) (19 21.7) ... (22 21.7) 0 73. X 74. a. 70 545 34.06 16 2116 X 70.5333 30 Median = 37.50 75. 370.08, found by 18,504/50 76. a. b. c. 77. Xw $5.00(270) $6.50(300) $8.00(100) $6.12 270 300 100 78. Xw 3(4) 3(4) 5(3) 2(3) 1(4) 3.50 3 3 5 2 1 79. Xw [15,300( 4.5) 10,400(3.0) 150,600(10.2)] 9.28 176,300 80. a. b. c. arithmetic mean = 6.49 km median = 6.3 km mode = 5.3 and 4.6 km (bimodal) 81. GM = 21 82. GM = 10 GM = 10 a. b. 55, found by 72 – 17 14.4, found by 144/10 where X = 43.2 83. 4.84, found by 121/25 Median = 4.0 On half the days she made at least 4 appointments. The arithmetic mean number of appointments per day is 4.84. 6,286,800 1 1.0094, so about 0.94% 5,164,900 33,598 1 1.0300 1 0.03 or 3.0 percent 25,000 44,771 1 1.0599995 1 0.06 or 6.0 percent 25,000 31 Chapter 3 c. 17.6245 84. a. b. c. 9, found by 12 – 3 2.72, found by 13.6/5 where mean = 7.6 3.5071 85. a. b. c. population 183.47 94.92% 86. a. 30 found by 30 – 0 (2094) 2 150 149 34,758 b. 6.09 found by 87. The following results were found using MINITAB Q1 = 44.25 Q3 = 68.50 Median = 55.50 The distribution is approximately symmetric. The box plot is as follows. -------------------------------------------| + |-----------------------------------|----------|----------|----------|----------|----------|--24 36 48 60 72 84 88. a. b. c. ---------------------+ ------------------------------------|-----|-----|-----|-----|-----|-----|-----| 10 15 20 25 30 35 40 45 No outliers The distribution is positively skewed. The median time to change a muffler is 23.50 minutes. The first quartile is 15.75 minutes and the third quartile is 29.25 minutes. The range of time is 10 minutes to 44 minutes. 89. The distribution is positively skewed. The first quartile is approximately $20 and the third quartile is approximately $90. There is one outlier located at $255. The median is about 50. 90. The distribution is positively skewed. The first quartile is equal to 10 and the third quartile is equal to 40. There are four outliers located at 85, 86, 95 and 99. The median is about 25. 91. a. b. c. d. Chapter 3 857.90 Median = 16.35 17158 . 50 (857.90)2 20,206.73 50 s 10.58 50 1 17158 . (2)(10.58) 4.002 and 38.318 10.58 CV (100) 61.66 % 17158 . X 32 e. f. 92. 3(17158 . 16.35) 0.23 10.58 25 75 Lp (50 1) 12.75 Lp (50 1) 38.25 100 100 Q1 = 7.825 Q3 = 27.400 ---------------------------| + |----------------------------------+----------+---------+----------+----------+----0 10 20 30 40 sk g. The distribution is nearly symmetrical. The mean is 17.158, the median is 16.35 and the standard deviation is 10.58. About 75 percent of the companies have a value less than 27.4 and 25 percent have a value less than 7.825. a. Computer Output: Variable Assets 5yr 1yr Variable Assets 5yr 1yr N 20 20 20 Mean 42220 145.8 -0.27 Minimum 22742 41.4 12.90 Median 36018 132.7 -1.60 Maximum 104357 264.3 13.20 TrMean 39850 145.0 -0.32 Q1 29097 119.8 -4.20 StDev 22495 48.8 6.10 SE Mean 5030 10.9 1.36 Q3 45484 184.9 2.88 Five-year returns are more variable. b. (22,495/42,220)(100) = 53.3% for assets; (48.8/145.8)(100) = 33.5% for five-year returns; (6.10/0.27)(100) = 2259.3% for one-year returns. The relative variation of one-year returns is quite high. c. Software gives the following coefficients of skewness: Assets 2.170968 Five-year 0.45416 One-year 0.484715 Assets are definitely positively skewed, but the other two variables are symmetric. d. See printout above. e. The box plot for five-year rates of return is: -------------------| + |---------------------------+----------+---------+-------50 150 250 The box plot for one-year rates of return is: 33 Chapter 3 ----------------| + |-------------------+----------+---------+-----10 0 10 93. a. The mean is 173.77 hours, found by 2259/13. The median is 195 hours. s = 105.61 hours, found by b. c. d. 2259 2 526,391 13 . 12 105.61 CV = 60.78%, found by 100 173.77 Coefficient of skewness is –0.697 L45 = (14)(.45) = 6.3. So the 45th percentile is 192 + 0.3(195192) = 192.9. L82 = (14)(.82) = 11.48. So the 82nd percentile is 260 + 0.48(295260) = 276.8. ----------------------| + |------------------------------+------+------+------+------+------+---0 75 150 225 300 375 There is a slight negative skewness visible, but no outliers. 94. a. b. 95. X 96.55 found by 10,620/110 s 1,029,937.5 109 (10,620) 2 110 6.514 Mean is 13, found by 910/70 s 96. a. b. 13,637.50 69 (910) 2 70 5118 . mean = $2706.67; median = $2235 BoxPlot 0 1000 2000 3000 4000 5000 6000 #1 There is one extreme outlier. This observation will distort the value of the mean. Chapter 3 34 7000 c. 97. The median will be more representative of the data than the mean due to the high extreme outlier. The following output is from MegaStat. a. Mean sample variance sample standard deviation Minimum Maximum Range 437,595.96 69,666,712,637 263,944.53 137000 1600000 1463000 Skewness Kurtosis coefficient of variation (CV) 1.52 4.14 60.32% 1st quartile Median 3rd quartile interquartile range Mode 200,000 435,000 574,000 374,000 200,000 1. see the above output. 2. see the above output. The distribution is positively skewed. 3. BoxPlot 0 200000 400000 600000 800000 1000000 1200000 1400000 1600000 1800000 List price There are two mild outliers. From the boxplot, the 1st and 3rd quartiles are about 200 000 and 575 000. 4. The outliers will distort the value of the mean, and so the median is more representative of the data. b. Count Mean sample variance sample standard deviation Minimum Maximum 99 2,096.91 1,447,318.23 1,203.05 196 10600 35 Chapter 3 Range 10404 Skewness Kurtosis coefficient of variation (CV) 1st quartile Median 3rd quartile interquartile range Mode 3.89 24.90 57.37% 1,356.00 1,946.00 2,479.50 1,123.50 1,400.00 1. see the above output 2. see the above output. The distribution is positively skewed. 3. BoxPlot 0 2000 4000 6000 8000 10000 12000 Size (in sq ft) There are two mild and one extreme outlier. From the boxplot, the 1st and 3rd quartiles are about 1400 and 2500. 4. The outliers will distort the value of the mean, and so the median is more representative of the data. 98. a. Chapter 3 1. 2. The mean is 65.49; median is 64.0 and the standard deviation is 24.87. The coefficient of skewness is positive, namely 0.289022. 36 3. There are no outliers. The first quartile is near 43 and the third quartile is about 86. 20 30 40 50 60 70 80 90 100 110 120 Salary b. 4. The distribution is fairly symmetric around 65. It seems to have large “shoulders” at the quartiles and then fall off very quickly. 1. 2. 3. The mean is 24.13; median = 14.0 and the standard deviation is 24.72. The coefficient of skewness is quite positive, namely 1.52649. There are two outliers at 88 and 90. The first quartile is near 6 and the third is 36. 0 10 20 30 40 50 60 70 80 90 Age c. 4. The age distribution is quite skewed with a typical value of 14 years and most of the stadiums are between 5 and 40 years of age. 1. 2. 3. The mean is 46,947; median = 46,056 and the standard deviation is 6475. The coefficient of skewness is positive, namely 0.330510. There are no outliers. The first quartile is near 42,000 and the third quartile is near 51,000. 37 Chapter 3 30000 40000 50000 60000 Size 4. 99. a. 1. 2. 3. The distribution is quite symmetric around 46,500 with the most values within 5000 of the center. The mean is 73.81; median = 76.10 and the standard deviation is 6.90 The distribution is quite negatively skewed with the coefficient of skewness equal to 2.10027. There are two outliers at 48 and 51. The first quartile is near 72 and the third quartile is near 78. 50 60 70 80 Life Expectancy 4. b. Chapter 3 The life expectancy distribution is heavily concentrated near 75 with a slight negative tail into the 60’s and , of course, the two outliers near 50. Select the variable GDP/cap 1. The mean is 16.58; median = 17.45 and the standard deviation = 9.27. 2. The distribution is fairly symmetric with the coefficient of skewness equal to 0.0567462. 3. There are no outliers. The first quartile is about 8 and the third quartile is near 24. 38 0 10 20 30 40 GDP/cap 4. The distribution of GDP/capita is symmetric about 16 and most of the values are within 8 of that center. 39 Chapter 3