Test bank and solution manual of mahtematich (2)

CHAPTER Descriptive Statistics 2.1 FREQUENCY DISTRIBUTIONS AND THEIR GRAPHS 2.1 Try It Yourself Solutions 1a The number of classes is b Min = 26, Max = 86, Class width = Range 86 - 26 = = 8.57  Number of classes c Lower limit 26 35 44 53 62 71 80 Upper limit 34 43 52 61 70 79 88 Class 26-34 35-43 44-52 53-61 62-70 71-79 80-88 Frequency, f 12 18 11 1 de 2a See part (b) b Class Frequency, f Midpoint 26-34 35-43 44-52 53-61 62-70 71-79 80-88 12 18 11 1 30 39 48 57 66 75 84 å f = 50 Relative frequency 0.04 0.10 0.24 0.36 0.22 0.02 0.02 f å n =1 Cumulative frequency 19 37 48 49 50 c Sample answer: The most common age bracket for the 50 most powerful women is 53-61 Eighty-six percent of the 50 most powerful women are older than 43 Four percent of the 50 most powerful women are younger than 35 15 Copyright © 2015 Pearson Education, Inc 16 CHAPTER DESCRIPTIVE STATISTICS 3a Class Boundaries 25.5-34.5 34.5-43.5 43.5-52.5 52.5-61.5 61.5-70.5 70.5-79.5 79.5-88.5 b Use class midpoints for the horizontal scale and frequency for the vertical scale (Class boundaries can also be used for the horizontal scale.) c d Same as 2(c) 4a Same as 3(b) bc d The frequency of ages increases up to 57 years old and then decreases 5abc Copyright © 2015 Pearson Education, Inc CHAPTER DESCRIPTIVE STATISTICS 17 6a Use upper class boundaries for the horizontal scale and cumulative frequency for the vertical scale bc Sample answer: The greatest increase in cumulative frequency occurs between 52.5 and 61.5 7a Enter data b 2.1 EXERCISE SOLUTIONS Organizing the data into a frequency distribution may make patterns within the data more evident Sometimes it is easier to identify patterns of a data set by looking at a graph of the frequency distribution If there are too few or too many classes, it may be difficult to detect patterns because the data are too condensed or too spread out Class limits determine which numbers can belong to that class Class boundaries are the numbers that separate classes without forming gaps between them Relative frequency of a class is the portion, or percentage, of the data that falls in that class Cumulative frequency of a class is the sum of the frequencies of that class and all previous classes The sum of the relative frequencies must be or 100% because it is the sum of all portions or percentages of the data A frequency polygon displays frequencies or relative frequencies whereas an ogive displays cumulative frequencies False Class width is the difference between the lower (or upper limits) of consecutive classes True Copyright © 2015 Pearson Education, Inc 18 CHAPTER DESCRIPTIVE STATISTICS False An ogive is a graph that displays cumulative frequencies 10 True Range 64 - = » 7.9  Number of classes Lower class limits: 9, 17, 25, 33, 41, 49, 57 Upper class limits: 16, 24, 32, 40, 48, 56, 64 11 Class width = Range 88 -12 = » 12.7  13 Number of classes Lower class limits: 12, 25, 38, 51, 64, 77 Upper class limits: 24, 37, 50, 63, 76, 89 12 Class width = Range 135 -17 = = 14.75  15 Number of classes Lower class limits: 17, 32, 47, 62, 77, 92, 107, 122 Upper class limits: 31, 46, 61, 76, 91, 106, 121, 136 13 Class width = Range 247 - 54 = = 19.3  20 Number of classes 10 Lower class limits: 54, 74, 94, 114, 134, 154, 174, 194, 214, 234 Upper class limits: 73, 93, 113, 133, 153, 173, 193, 213, 233, 253 14 Class width = 15 (a) Class width = 31 - 20 = 11 (b) and (c) Class 20-30 31-41 42-52 53-63 64-74 75-85 86-96 Midpoint 25 36 47 58 69 80 91 Class boundaries 19.5-30.5 30.5-41.5 41.5-52.5 52.5-63.5 63.5-74.5 74.5-85.5 85.5-96.5 16 (a) Class width = 10 - = 10 (b) and (c) Class 0-9 10-19 20-29 30-39 40-49 50-59 60-69 Midpoint 4.5 14.5 24.5 34.5 44.5 54.5 64.5 Copyright © 2015 Pearson Education, Inc Class boundaries -0.5 -9.5 9.5-19.5 19.5-29.5 29.5-39.5 39.5-49.5 49.5-59.5 59.5-69.5 CHAPTER DESCRIPTIVE STATISTICS 17 Class Frequency, f Midpoint 20-30 31-41 42-52 53-63 64-74 75-85 86-96 19 43 68 69 74 68 24 25 36 47 58 69 80 91 å f = 365 Relative frequency 0.05 0.12 0.19 0.19 0.20 0.19 0.07 f å n »1 Cumulative frequency 19 62 130 199 273 341 365 Relative frequency 0.15 0.30 0.22 0.17 0.07 0.06 0.03 f å n =1 Cumulative frequency 188 560 824 1029 1112 1188 1220 18 Class Frequency, f Midpoint 0-9 10-19 20-29 30-39 40-49 50-59 60-69 188 372 264 205 83 76 32 4.5 14.5 24.5 34.5 44.5 54.5 64.5 å f = 1220 19 (a) Number of classes = (c) Greatest frequency ≈ 300 (b) Least frequency ≈ 10 (d) Class width = 10 20 (a) Number of classes = (c) Greatest frequency = 23 (b) Least frequency = (d) Class width = 53 21 (a) 50 (b) 345.5-365.5 pounds 22 (a) 50 (b) 64-66 inches 23 (a) 15 (c) 31 – = 25 (b) 385.5 pounds (d) 50 – 42 = 24 (a) 48 (c) 25 – = 20 (b) 66 inches (d) 50 – 44 = 25 (a) Class with greatest relative frequency: 39-40 centimeters Class with least relative frequency: 34-35 centimeters (b) Greatest relative frequency ≈ 0.25 Least relative frequency ≈ 0.02 (c) Approximately 0.08 Copyright © 2015 Pearson Education, Inc 19 20 CHAPTER DESCRIPTIVE STATISTICS 26 (a) Class with greatest relative frequency: 19-20 minutes Class with least relative frequency: 21-22 minutes (b) Greatest relative frequency ≈ 40% Least relative frequency ≈ 2% (c) Approximately 33% 27 Class with greatest frequency: 29.5-32.5 Classes with least frequency: 11.5-14.5 and 38.5-41.5 28 Class with greatest frequency: 7.75-8.25 Class with least frequency: 6.25-6.75 Range 39 - = = 7.8  Number of classes Class Frequency, f Midpoint Relative frequency 0-7 3.5 0.32 8-15 11.5 0.32 16-23 19.5 0.12 24-31 27.5 0.12 32-39 35.5 0.12 f å f = 25 å n =1 Classes with greatest frequency: 0-7, 8-15 Classes with least frequency: 16-23, 24-31, 32-39 29 Class width = Range 530 - 30 = » 83.3  84 Number of classes Class Frequency, f Midpoint Relative frequency 30-113 71.5 0.1724 114-197 155.5 0.2414 198-281 239.5 0.2759 282-365 323.5 0.0690 366-449 407.5 0.1034 450-533 491.5 0.1379 f å f = 29 å n =1 Class with greatest frequency: 198-281 Class with least frequency: 282-365 Cumulative frequency 16 19 22 25 30 Class width = Copyright © 2015 Pearson Education, Inc Cumulative frequency 12 20 22 25 29 CHAPTER 31 Class width = Class 1000-2019 2020-3039 3040-4059 4060-5079 5080-6099 6100-7119 DESCRIPTIVE STATISTICS Range 7119 -1000 = » 1019.8  1020 Number of classes Frequency, f Midpoint Relative Cumulative frequency frequency 12 1509.5 0.5455 12 2529.5 0.1364 15 3549.5 0.0909 17 4569.5 0.1364 20 5589.5 0.0455 21 6609.5 0.0455 22 f å f = 22 å n »1 Sample answer: The graph shows that most of the sales representatives at the company sold between $1000 and $2019 32 Class width = Class 32-35 36-39 40-43 44-47 48-51 Range 51 - 32 = = 3.8  Number of classes Frequency, f Midpoint Relative frequency 33.5 0.1250 37.5 0.3750 41.5 0.3333 45.5 0.1250 49.5 0.0417 f å f = 24 å n =1 Cumulative frequency 12 20 23 24 Sample answer: The graph shows that most of the pungencies of the peppers were between 36,000 and 43,000 Scoville units Copyright © 2015 Pearson Education, Inc 21 22 CHAPTER 33 Class width = Class 291-318 319-346 347-374 375-402 403-430 431-458 459-486 487-514 DESCRIPTIVE STATISTICS Range 514 - 291 = = 27.875  28 Number of classes Frequency, f Midpoint Relative frequency 304.5 0.1667 332.5 0.1333 360.5 0.1000 388.5 0.1667 416.5 0.2000 444.5 0.1333 472.5 0.0333 500.5 0.0667 f å f = 30 å n =1 Cumulative frequency 12 17 23 27 28 30 Sample answer: The graph shows that the most frequent reaction times were between 403 and 430 milliseconds 34 Class width = Class 1250-1380 1381-1511 1512-1642 1643-1773 1774-1904 1905-2035 2036-2166 2167-2297 Range 2296 -1250 = = 130.75  131 Number of classes Frequency, f Midpoint Relative Cumulative frequency frequency 1315 0.10 1446 0.15 1577 0.30 11 1708 0.05 12 1839 0.20 16 1970 0.10 18 2101 0.05 19 2232 0.05 20 f å f = 20 å n =1 Copyright © 2015 Pearson Education, Inc CHAPTER DESCRIPTIVE STATISTICS 23 Sample answer: The graph shows that the most frequent finishing times were from 1381 to 1642 seconds and from 1774 to 2035 seconds 35 Class width = Class 1-2 3-4 5-6 7-8 9-10 Range 10 -1 = = 1.8  Number of classes Frequency, f Midpoint Relative frequency 1.5 0.0833 3.5 0.0833 5.5 0.2083 10 7.5 0.4167 9.5 0.2083 f å f = 24 å n »1 Cumulative frequency 19 24 Class with greatest relative frequency: 7-8 Class with least relative frequency: 1-2 and 3-4 36 Class width = Class 6-7 8-9 10-11 12-13 14-15 Range 14 - = = 1.6  Number of classes Frequency, f Midpoint Relative frequency 6.5 0.12 10 8.5 0.38 10.5 0.23 12.5 0.23 14.5 0.04 f å f = 26 å n =1 Copyright © 2015 Pearson Education, Inc Cumulative frequency 13 19 25 26 24 CHAPTER DESCRIPTIVE STATISTICS Class with greatest relative frequency: 8-9 Class with least relative frequency: 14-15 37 Class width = Class 417-443 444-470 471-497 498-524 525-551 Range 547 - 417 = = 26  27 Number of classes Frequency, f Midpoint Relative frequency 430 0.20 457 0.20 484 0.24 511 0.16 538 0.20 f å f = 25 å n =1 Cumulative frequency 10 16 20 25 Class with greatest relative frequency: 471-497 Class with least relative frequency: 498-524 38 Class width = Class 138-202 203-267 268-332 333-397 398-462 Range 462 -138 = = 64.8  65 Number of classes Frequency, f Midpoint Relative frequency 12 170 0.46 235 0.23 300 0.15 365 0.04 430 0.12 f å f = 26 å n =1 Copyright © 2015 Pearson Education, Inc Cumulative frequency 12 18 22 23 26 42 CHAPTER DESCRIPTIVE STATISTICS 10 The shape of the distribution is symmetric because a vertical line can be drawn down the middle, creating two halves that are approximately the same 11 The shape of the distribution is uniform because the bars are approximately the same height 12 The shape of the distribution is skewed left because the bars have a “tail” to the left 13 (11), because the distribution values range from to 12 and has (approximately) equal frequencies 14 (9), because the distribution has values in the thousands of dollars and is skewed right due to the few executives that make a much higher salary than the majority of the employees 15 (12), because the distribution has a maximum value of 90 and is skewed left due to a few students scoring much lower than the majority of the students 16 (10), because the distribution is approximately symmetric and the weights range from 80 to 160 pounds å x 192 = » 14.8 13 n 12 12 13 14 14 15 15 15 16 16 16 16 18 median = 15 mode = 16 (occurs times) 17 x = å x 1205 = » 172.1 n 169 169 170 172 174 175 176 median = 172 mode = 169 (occurs times) The mode does not represent the center of the data because 169 is the smallest number in the data set 18 x = å x 8249 = » 1178.4 n 818 1125 1155 1229 1275 1277 1370 median = 1229 mode = none The mode cannot be found because no data entry is repeated 19 x = å x 414 = = 46 n 36 38 40 43 43 49 50 52 63 median = 43 mode = 43 (occurs times) 20 x = Copyright © 2015 Pearson Education, Inc CHAPTER DESCRIPTIVE STATISTICS 43 å x 603 = » 43.1 14 n 39 40 41 42 42 42 44 44 44 44 44 45 45 47  21 x = median = 44 + 44 = 44 mode = 44 (occurs times) å x 2004 = = 200.4 10 n 154 171 173 181 184 188 203 235 240 275    22 x = 184 + 188 = 186 mode = none; The mode cannot be found because no data entry is repeated å x 481 23 x = = » 30.1 16 n 17 21 21 25 26 30 31 31 31 34 34 34 35 36 37 38  median = 31 + 31 = 31 mode = 31 and 34 (both occur times) median = å x 1223 = » 61.2 20 n 12 18 26 28 31 33 40 44 45 49 61 63 75 80 80 89 96 103 125 125  24 x = median = 49 + 61 = 55 mode = 80, 125 The modes not represent the center of the data set because they are large values compared to the rest of the data å x 83 = = 16.6 n 1.0 10.0 15.0 25.5 31.5 median = 43 mode = none The mode cannot be found because no data entry is repeated 25 x = å x 197.5 = » 14.11 14 n 1.5 2.5 2.5 5.0 10.5 11.0 13.0 15.5 16.5 17.5 20.0 26.5 27.0 28.5   26 x = median = mode = 2.5 (occurs times) Copyright © 2015 Pearson Education, Inc 13.0 + 15.5 = 14.25 44 CHAPTER DESCRIPTIVE STATISTICS The mode does not represent the center of the data set because 2.5 is much smaller than most of the data in the set 27 x is not possible (nominal data) median = not possible (nominal data) mode = “Eyeglasses” The mean and median cannot be found because the data are at the nominal level of measurement 28 x is not possible (nominal data) median is not possible (nominal data) mode = “Money needed” The mean and median cannot be found because the data are at the nominal level of measurement 29 x is not possible (nominal data) median is not possible (nominal data) mode = “Junior” The mean and median cannot be found because the data are at the nominal level of measurement 30 x is not possible (nominal data) median is not possible (nominal data) mode = “on Facebook, find it valuable” The mean and median cannot be found because the data are at the nominal level of measurement å x 835 = » 29.8 28 n 12 15 18 19 20 24 24 24 25 28 29 32 32 33 35 35 36 38 39 40 41 42 47 48 51  31 x = median = 32 + 32 = 32 mode = 24, 35 (both occur times each) å x 29.9 = » 2.49 12 n 0.8 1.5 1.6 1.8 2.1 2.3 2.4 2.5 3.0 3.9 4.0 4.0    32 x = median = 2.3 + 2.4 = 2.35 mode = 4.0 (occurs times) The mode does not represent the center of the data set because it is the largest value in the data set å x 292 = » 19.5 15 n 10 15 15 15 17 20 21 22 22 25 28 32 37 median = 20 mode = 15 (occurs times) 33 x = Copyright © 2015 Pearson Education, Inc CHAPTER DESCRIPTIVE STATISTICS å x 3110 = » 207.3 15 n 170 180 190 200 200 210 210 210 210 210 220 220 220 220 240 median = 210 mode = 210 (occurs times) 34 x = 35 The data are skewed right A = mode, because it is the data entry that occurred most often B = median, because the median is to the left of the mean in a skewed right distribution C = mean, because the mean is to the right of the median in a skewed right distribution 36 The data are skewed left A = mean, because the mean is to the left of the median in a skewed left distribution B = median, because the median is to the right of the mean in a skewed left distribution C = mode, because it is the data entry that occurred most often 37 Mode, because the data are at the nominal level of measurement 38 Mean, because the data are symmetric 39 Mean, because the distribution is symmetric and there are no outliers 40 Median, because there is an outlier 41 Source Homework Quiz Project Speech Final exam x= å ( x ⋅ w) åw Score, x 85 80 100 90 93 = Weight, w 0.05 0.35 0.20 0.15 0.25 å w =1 x·w 4.25 28 20 13.5 23.25 å ( x ⋅ w) = 89 89 = 89 42 Source Article reviews Quizzes Midterm exam Student lecture Final exam x= å ( x ⋅ w) åw = Score, x 95 100 89 100 92 93 = 93 Copyright © 2015 Pearson Education, Inc Weight, w 10% 10% 30% 10% 40% å w = 100% x·w 9.5 10 26.7 10 36.8 å ( x ⋅ w) = 93 45 46 CHAPTER DESCRIPTIVE STATISTICS Balance, x $523 $2415 $250 Days, w 24 å w = 30 43 x= å ( x ⋅ w) åw = x·w 12,552 4830 1000 å ( x ⋅ w) = 18,382 18,382 » $612.73 30 44 Balance, x $759 $1985 $1410 $348 x= å ( x ⋅ w) åw Days, w 15 5 å w = 31 = x·w 11,385 9925 7050 2088 å ( x ⋅ w) = 30, 448 30, 448 » $982.20 31 45 Grade A B B C D x= å ( x ⋅ w) åw Points, x 3 = Credits, w 3 w å = 15 x·w 16 9 x ⋅ ( å w) = 42 42 = 2.8 15 46 Source Engineering Business Math x= å ( x ⋅ w) åw Score, x 85 81 90 = 2268 = 84 27 Copyright © 2015 Pearson Education, Inc Weight, w 13 w å = 27 x·w 765 1053 450 x ⋅ å ( w) = 2268 CHAPTER DESCRIPTIVE STATISTICS 47 Source Homework Quiz Project Speech Final exam x= å ( x ⋅ w) åw Score, x 85 80 100 90 85 = Weight, w 0.05 0.35 0.20 0.15 0.25 å w =1 x·w 4.25 28 20 13.5 21.25 å ( x ⋅ w) = 87 Credits, w 3 w å = 15 x·w 16 12 x ⋅ å ( w) = 45 87 = 87 48 Grade A A B C D x= Points, x 4 å ( x ⋅ w) åw = 45 =3 15 49 Class 29-33 34-38 39-43 44-48 x= Midpoint, x 31 36 41 46 Frequency, f 11 12 n = 30 x·f 341 432 82 230 å ( x ⋅ f ) = 1085 å ( x ⋅ f ) 1085 = » 36.2 miles per gallon n 30 50 Class 22-27 28-33 34-39 40-45 46-51 x= Midpoint, x 24.5 30.5 36.5 42.5 48.5 å(x ⋅ f ) n = Frequency, f 16 2 n = 24 702 » 29.3 miles per gallon 24 Copyright © 2015 Pearson Education, Inc x·f 392 61 73 127.5 48.5 å ( x ⋅ f ) = 702 47 48 CHAPTER DESCRIPTIVE STATISTICS 51 Class 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 x= Midpoint, x 4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 Frequency, f 44 66 32 53 35 31 23 13 n = 299 x·f 198.0 957.0 784.0 1828.5 1557.5 1689.5 1483.5 968.5 169.0 å ( x ⋅ f ) = 9,635.5 å ( x ⋅ f ) 9,635.5 = » 32.2 years old n 299 52 Class 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 x= Midpoint, x 4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 å(x ⋅ f ) n = 53 Class width = Class 127-161 162-196 197-231 232-266 267-301 Frequency, f 30 28 17 22 23 46 37 18 n = 225 x·f 135.0 406.0 416.5 759.0 1023.5 2507.0 2386.5 1341.0 338.0 x ⋅ ( å f ) = 9312.5 9312.5 » 41.4 years old 225 Range 297 -127 = = 34  35 Number of classes Midpoint Frequency, f 144 179 214 249 284 å f = 24 Shape: Positively skewed Copyright © 2015 Pearson Education, Inc CHAPTER 54 Class width = Class 3-4 5-6 7-8 9-10 11-12 13-14 Range 14 - = » 1.83  Number of classes Midpoint Frequency, f 3.5 5.5 7.5 9.5 11.5 13.5 å f = 20 Shape: Positively skewed 55 Class width = Class 62-64 65-67 68-70 71-73 74-76 Range 76 - 62 = = 2.8  Number of classes Midpoint Frequency, f 63 66 69 72 75 å f = 30 Shape: Symmetric Copyright © 2015 Pearson Education, Inc DESCRIPTIVE STATISTICS 49 50 CHAPTER 56 Class width = Class DESCRIPTIVE STATISTICS Range -1 = = 0.8333  Number of classes Frequency, f 6 å f = 30 Shape: Uniform å x 36.03 = = 6.005 n 5.59 5.99 6.00 6.02 6.03 6.40   57 (a) x = median = 6.00 + 6.02 = 6.01 å x 35.67 = = 5.945 n 5.59 5.99 6.00 6.02 6.03 6.04   (b) x = median = 6.00 + 6.02 = 6.01 (c) The mean was affected more å x 1104.3 = » 58.12 19 n 10.9 13.8 14.2 16.0 24.4 25.9 27.5 27.8 29.9 31.2 42.4 42.9 43.4 49.2 55.9 65.7 103.9 198.4 280.9 median = 31.2 å x 823.4 (b) x = = » 45.74 18 n 58 (a) x = Copyright © 2015 Pearson Education, Inc CHAPTER DESCRIPTIVE STATISTICS 51 10.9 13.8 14.2 16.0 24.4 25.9 27.5 27.8 29.9 31.2 42.4 42.9 43.4 49.2 55.9 65.7 103.9 198.4   median = 29.9 + 31.2 = 30.55 The mean was affected more (c) x = å x 1125.8 = = 56.29 20 n 10.9 13.8 14.2 16.0 21.5 24.4 25.9 27.5 27.8 29.9 31.2 42.4 42.9 43.4 49.2 55.9 65.7 103.9 198.4 280.9   median = 29.9 + 31.2 = 30.55 The mean was affected more 59 Clusters around 16-21 and around 36 60 Cluster around 18-27, gap between 27 and 72, outlier at 72 61 Sample answer: Option 2; The two clusters represent different types of vehicles which can be more meaningfully analyzed separately For instance, suppose the mean gas mileage for cars is very far from the mean gas mileage for trucks, vans, and SUVs Then, the mean gas mileage for all of the vehicles would be somewhere in the middle and would not accurately represent the gas mileages of either group of vehicles å x 3222 = = 358 n 147 177 336 360 375 393 408 504 522 median = 375 å x 9666 (b) x = = = 1074 n 441 531 1008 1080 1125 1179 1224 1512 1566 median = 1125 (c) The mean and median in part (b) are three times the mean and median in part (a) 62 (a) x = (d) If you multiply the mean and median of the original data set by 36, you will get the mean and median of the data set in inches 63 Car A å x 152 x= = = 30.4 n 28 28 30 32 34 median = 30 mode = 28 (occurs times) Copyright © 2015 Pearson Education, Inc 52 CHAPTER DESCRIPTIVE STATISTICS Car B å x 151 = = 30.2 x= n 29 29 31 31 31 median = 31 mode = 31 (occurs times) Car C å x 151 x= = = 30.2 n 28 29 30 32 32 median = 30 mode = 32 (occurs times) (a) Mean should be used because Car A has the highest mean of the three (b) Median should be used because Car B has the highest median of the three (c) Mode should be used because Car C has the highest mode of the three 34 + 28 = 31 31 + 29 = 30 Car B: Midrange = 32 + 28 = 30 Car C: Midrange = Car A because the midrange is the largest 64 Car A: Midrange = å x 1477 = » 49.2 30 n 11 13 22 28 36 36 36 37 37 37 38 41 43 44 46 47 51 51 51 53 61 62 63 64  65 (a) x = 72 72 74 76 85 90 (b) Key: = 36 11 median 22 36 6 7 41 mean 51 1 61 72 85 90 (c) The distribution is positively skewed Copyright © 2015 Pearson Education, Inc median = 46 + 47 = 46.5 CHAPTER DESCRIPTIVE STATISTICS 53 66 (a) Order the data values 11 13 22 28 36 36 36 37 37 37 38 41 43 44 46 47 51 51 51 53 61 62 63 64 72 72 74 76 85 90 Delete the lowest 10%, smallest observations (11, 13, 22) Delete the highest 10%, largest observations (76, 85, 90) Find the 10% trimmed mean using the remaining 24 observations å x 1180 x= = » 49.2 24 n 10% trimmed mean » 49.2 (b) x » 49.2 median = 46.5 mode = 36, 37, 51 90 + 11 midrange = = 50.5 (c) Using a trimmed mean eliminates potential outliers that may affect the mean of all the observations 2.4 MEASURES OF VARIATION 2.4 Try It Yourself Solutions 1a Min = 23, or $23,000 and Max = 58, or $58,000 b Range = Max – Min = 58 – 23 = 35, or $35,000 c The range of the starting salaries for Corporation B is 35, or $35,000 This is much larger than the range for Corporation A 2ab m = 41.5 , or $41,500 Salary, x x– 23 –18.5 29 –12.5 32 –9.5 40 –1.5 41 –0.5 41 –0.5 49 7.5 50 8.5 52 10.5 58 16.5 å x = 415 å ( x - m) = å ( x - m) (x – )2 342.25 156.25 90.25 2.25 0.25 0.25 56.25 72.25 110.25 272.25 å ( x - m) = 1102.5 1102.5 » 110.3 10 1102.5 d s = s = = 10.5, or $10,500 10 e The population variance is about 110.3 and the population standard deviation is 10.5, or $10,500 c s = N = Copyright © 2015 Pearson Education, Inc 54 CHAPTER 3ab x = DESCRIPTIVE STATISTICS å x 316 = = 39.5 n Time, x x-x 43 57 18 45 47 33 49 24 å x = 316 3.5 17.5 -21.5 5.5 7.5 -6.5 9.5 -15.5 å ( x - m) = ( x - x) 12.25 306.25 462.25 30.25 56.25 42.25 90.25 240.25 å ( x - m) = 1240 SS x = å ( x - x) = 1240 å ( x - x) 1240 = » 177.1 n -1 1240 c s = s = » 13.3 b s = 4a Enter the data in a computer or a calculator b x » 22.1, s » 5.3 5a Sample answer: 7, 7, 7, 7, 7, 13, 13, 13, 13, 13 b Salary, x x– (x – )2 –3 –3 –3 –3 –3 13 13 13 13 13 å x = 100 å ( x - m) = å ( x - m) = 90 m= å x 100 = = 10 N 10 å ( x - m) s= N = 90 = =3 10 6a 67.1 – 64.2 = 2.9 = standard deviation b 34% c Approximately 34% of women ages 20-29 are between 64.2 and 67.1 inches tall Copyright © 2015 Pearson Education, Inc CHAPTER DESCRIPTIVE STATISTICS 7a 35.3 – 2(21.1) = -6.9 Because –6.9 does not make sense for an age, use b 35.3 + 2(21.1) = 77.5 1 c - = - = - = 0.75 k ( 2) At least 75% of the data lie within standard deviations of the mean At least 75% of the population of Alaska is between and 77.5 years old 8a x b x = f 10 19 7 1 n = 50 xf 19 14 21 20 å xf = 85 å xf 85 = = 1.7 n 50 c x- x ( x - x) ( x - x) –1.7 –0.7 0.3 1.3 2.3 3.3 4.3 2.89 0.49 0.09 1.69 5.29 10.89 18.49 28.90 9.31 0.63 11.83 26.45 10.89 18.49 2 å ( x - x) å ( x - x) f f f = 106.5 d s = n -1 = 106.5 » 1.5 49 9a Class 1-99 100-199 200-299 300-399 400-499 500+ b x = x 49.5 149.5 249.5 349.5 449.5 650 f 380 230 210 50 60 70 n = 1000 å xf 195,535 = » 195.5 1000 n Copyright © 2015 Pearson Education, Inc xf 18,810 34,385 52,395 17,475 26,970 45,500 å xf = 195,535 55 56 CHAPTER DESCRIPTIVE STATISTICS c x- x ( x - x) –146.0 –46.0 54.0 154.0 254.0 454.5 21,316 2116 2916 23,716 64,516 206,570.25 ( x - x) 2 8,100,080 486,680 612,360 1,185,800 3,870,960 14,459,917.5 å ( x - x) å ( x - x) f f f = 28,715,797.5 d s = n -1 = 28,715,797.5 » 169.5 999 10a Los Angeles: x » 31.0 , s » 12.6 Dallas/Fort Worth: x » 22.1 , s » 5.3 s 12.6 ⋅100% » 40.6% b Los Angeles: CV = = x 31.0 5.3 s Dallas/Fort Worth: CV = = ⋅100% » 24.0% x 22.1 c The office rental rates are more variable in Los Angeles than in Dallas/Fort Worth 2.4 EXERCISE SOLUTIONS The range is the difference between the maximum and minimum values of a data set The advantage of the range is that it is easy to calculate The disadvantage is that it uses only two entries from the data set A deviation ( x - m ) is the difference between an entry x and the mean of the data The sum of the deviations is always zero The units of variance are squared Its units are meaningless (example: dollars2) The units of standard deviation are the same as the data The standard deviation is the positive square root of the variance The standard deviation and variance can never be negative because squared deviations can never be negative When calculating the population standard deviation, you divide the sum of the squared deviations by N, then take the square root of that value When calculating the sample standard deviation, you divide the sum of the squared deviations by n - , then take the square root of that value When given a data set, you would have to determine if it represented the population or if it was a sample taken from the population If the data are a population, then s is calculated If the data are a sample, then s is calculated Copyright © 2015 Pearson Education, Inc ... frequency of a class is the portion, or percentage, of the data that falls in that class Cumulative frequency of a class is the sum of the frequencies of that class and all previous classes The sum of. .. 62%; The proportion of scores greater than or equal to 1610 is 0.62 (c) A score of 1357 or above, because the sum of the relative frequencies of the class starting with 1357 and all classes with... 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