Mann - Introductory Statistics, Fifth Edition, Solutions Manual

Chapter Two2.1 Data in their original form are usually too large and unmanageable. By grouping data, we make them manageable. It is easier to make decisions and draw conclusions using grouped data than ungrouped data.2.2 The relative frequency for a category is obtained by dividing the frequency of that category by the sum of the frequencies of all categories. The percentage for a category is obtained by multiplying the relative frequency of that category by 100. Exercise 2.3 can be considered as an example to show how relative frequencies and percentages are calculated.2.3 a. & b. Category Frequency Relative Frequency Percentage A 8 .267 26.7 B 8 .267 26.7 C 14 .467 46.7 c. 26.7 % of the elements in this sample belong to category B. d. 26.7% + 46.7% = 73.4% of the elements in this sample belong to category A or C. e. 16 y c 12 n e u 8 q e r 4 F 0 A B C Category2.4 a. & b. Category Frequency Relative Frequency Percentage Y 23 .575 57.5 N 13 .325 32.5 D 4 .100 10.0 c. 57.5% of the elements belong to category Y. d. 32.5 + 10 = 42.5% of the elements belong to categories N or D.9 10 e. D 1 0 . 0 %N Y 3 2 . 5 % 5 7 . 5 %2.5 a. & b. Category Frequency Relative Frequency Percentage F 12 .24 24 SO 12 .24 24 J 15 .30 30 SE 11 .22 22 c. 30 + 22 = 52% of the students are juniors or seniors. d.16 y c 12 n e u 8 q e r 4 F 0 F SO J SE Category2.6 a. & b. Category Frequency Relative Frequency Percentage T 4 .133 13.3 R 10 .333 33.3 A 7 .233 23.3 P 8 .267 26.7 M 1 .033 3.3 c. 33.3 + 23.3 = 56.6% of the adults ranked refrigerators or air conditioning as the convenience they would find the most difficult to do without. d.0.40 y c e 0.30 v n i e t a u 0.20 l q e e R r 0.10 F 0.00 T R A P M Category Mann - Introductory Statistics , Fifth Edition, Solutions Manual 112.7 a. & b. Category Frequency Relative Frequency Percentage C 9 9/20= .45 45 F 5 5/20= .25 25 T 6 6/20= .30 30 c. 45% of the employees would prefer a four-day work week. d.T 30% C 45%F 25%2.8 a. & b. Category Frequency Relative Frequency Percentage C 4 .250 25.0 CK 5 .313 31.3 CC 4 .250 25.0 D 2 .125 12.5 O 1 .063 6.3 c.O 6 . 3 % D C 1 2 . 5 % 2 5 . 0 %C C 2 5 . 0 % C K 3 1 . 3 %2.9 Let the seven categories listed in the table be denoted by S, HC, R, O, E, P, and U respectively.P U 3% 8% E S 12% 26%HC O R 20% 12% 19%2.10 Let the four categories listed in the table be denoted by E, D, B, and O respectively. 12 y c n 50 e u 40 q e 30 rF 20 e 10 v i t 0 a l e E D B O R Category2.11 1. The number of classes to be used to group the given data. 2. The width of each class. 3. The lower limit of the first class. 2.12 The relative frequency for a class is obtained by dividing the frequency of that class by the sum of frequencies of all classes. The percentage for a class is obtained by multiplying the relative frequency of that class by 100. Exercise 2.18 can be considered as an example to illustrate the calculation of relative frequencies and percentages.2.13 A data set that does not contain fractional values is usually grouped by using classes with limits. Suppose we have data on ages of 100 managers, and ages are rounded to years. Then, the following table could be an example of grouped data that uses classes with limits.Ages (years) Frequency 21 to 30 12 31 to 40 27 41 to 50 31 51 to 60 22 61 to 70 8A data set that contains fractional values is grouped by using the less than method. Suppose we have data on sales of 100 medium sized companies. The following table shows a frequency table for such data.Sales(millions of dollars) Frequency 0 to less than 10 27 10 to less than 20 31 20 to less than 30 19 30 to less than 40 14 40 to less than 50 9Single valued classes are used to group a data set that contains only a few distinct values. As an example, suppose we have a data set on the number of children for 100 families. The following table is an example of a frequency table using single valued classes. Mann - Introductory Statistics , Fifth Edition, Solutions Manual 13Number of Children Frequency 0 13 1 26 2 38 3 18 4 52.14 a. & c. Class Boundaries Class Midpoint Relative Frequency Percentage -0.5 to less than 99.5 49.5 .39 39 99.5 to less than 199.5 149.5 .21 21 199.5 to less than 299.5 249.5 .18 18 299.5 to less than 399.5 349.5 .15 15 399.5 to less than 499.5 449.5 .07 7 b. Yes, each class has a width of 100. d. 18 + 15 + 7 = 40% of the students wrote 200 or more checks during 2002.2.15 a. & c. Class Boundaries Class Midpoint Relative Frequency Percentage 17.5 to less than 30.5 24 .24 24 30.5 to less than 43.5 37 .38 38 43.5 to less than 56.5 50 .28 28 56.5 to less than 69.5 63 .10 10 b. Yes, each class has a width of 13. d. 24 + 38 = 62% of the employees are 43 years old or younger.2.16 a. & b. Class Limits Class Boundaries Class Midpoints 1 to 200 .5 to less than 200.5 100.5 201 to 400 200.5 to less than 400.5 300.5 401 to 600 400.5 to less than 600.5 500.5 601 to 800 600.5 to less than 800.5 700.5 801 to 1000 800.5 to less than 1000.5 900.5 1001 to 1200 1000.5 to less than 1200.5 1100.52.17 a., b., & c. Class Limits Class Boundaries Class Width Class Midpoint 1 to 25 .5 to less than 25.5 25 13 26 to 50 25.5 to less than 50.5 25 38 51 to 75 50.5 to less than 75.5 25 63 76 to 100 75.5 to less than 100.5 25 88 101 to 125 100.5 to less than 125.5 25 113 126 to 150 125.5 to less than 150.5 25 138 14 2.18 a. & b. Team Revenue Frequency Relative Frequency Percentage 125 to 130 2 .133 13.3 131 to 136 5 .333 33.3 137 to 142 4 .267 26.7 143 to 148 2 .133 13.3 149 to 154 0 .000 0 155 to 160 2 .133 13.3 c. The data are skewed to the right. d. 26.7 + 13.3 + 0 +13.3 = 53.3% of these teams have revenue of $137 million or more.2.19 a. & b. Number of Computer Frequency Relative Frequency Percentage Terminals Manufactured 21 - 23 7 .233 23.3 24 - 26 4 .133 13.3 27 - 29 9 .300 30.0 30 - 32 4 .133 13.3 33 - 35 6 .200 20.0 c. e 40 g a 30 t n 20 e c r 10 eP 0 21- 24- 27- 30- 33- 23 26 29 32 35 Number of Computer Terminals d. For 30 % of the days, the number of computer terminals manufactured is in the interval 27 to 29.2.20 a. Keyboards Assembled Frequency Relative Frequency 41 – 44 5 .20 45 – 48 8 .32 49 – 52 8 .32 53 – 56 4 .16 b. & c.40 y c e v n 30 i e t a u 20 l q e e 10 R rF 0 41-44 45-48 49-52 53-56 Number of Keyboards Assembled Mann - Introductory Statistics , Fifth Edition, Solutions Manual 152.21 a. & b. Median Income Frequency Relative Frequency Percentage (thousands of dollars) 30 – 35 4 .211 21.1 35 – 40 7 .368 36.8 40 – 45 2 .105 10.5 45 – 50 4 .211 21.1 50 – 55 2 .105 10.52.22 a., b., & c. The minimum rent paid in the sample is $540, and the maximum rent paid is $1306. One possible answer to this question is given in the following table. However, we can obtain a different frequency distribution if we choose different classes.Apartment Rent (dollars) Frequency Relative Frequency Percentage Class Midpoint 540 – 694 11 .44 44 617 695 – 849 7 .28 28 772 850 – 1004 2 .08 8 927 1005 – 1159 2 .08 8 1082 1160 – 1314 3 .12 12 12372.23 a., b., & c. The lowest phone bill in our data is $16.99 and the highest amount is $33.71. The following table provides one way to group these data into five classes of equal width. Telephone Bill (dollars) Frequency Relative Frequency Percentage Class Midpoint 15 to less than 19 4 .16 16 17 19 to less than 23 11 .44 44 21 23 to less than 27 7 .28 28 25 27 to less than 31 2 .08 8 29 31 to less than 35 1 .04 4 332.24 a. & b. The minimum price of gasoline in this data set is $1.299, and the maximum price is $1.643. One way to group these data into four classes is shown below.Gasoline Prices (dollars) Frequency Relative Frequency Percentage 1.299 – 1.398 12 .48 48 1.399 – 1.498 8 .32 32 1.499 – 1.598 3 .12 12 1.599 – 1.698 2 .08 8 c. 60 y c 50 e v n i 40 e t u a 30 l q e 20 e r R 10 F 0 - - - - 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 3 4 5 6 2 5 3 4 ...... 1 1 1 1 1 1 1 1 Gasoline Prices 16 2.25 a. & b. The lowest cost of a visit to the doctor in our data is $52.33, and the highest cost is $88.67. One way to group these data using four classes is shown in the table here. However, your answer can be different depending on the number of classes used to group the data and the class width used.Visit to Doctor (dollars) Frequency Relative Frequency Percentage 50 to less than 60 9 .36 36 60 to less than 70 7 .28 28 70 to less than 80 7 .28 28 80 to less than 90 2 .08 8 c. 40 e 30 g a t n 20 e c r e 10 P 0 50-60 60-70 70-80 80-90 Price of Doctor Visits 2.26 a. & b. Price of Beer (dollars) Frequency Relative Frequency Percentage 6.30 – 6.90 4 .16 16 6.90 – 7.50 13 .52 52 7.50 – 8.10 6 .24 24 8.10 – 8.70 2 .08 82.27 a. & b ERA Frequency Relative Frequency Percentage 3.00 to less than 3.50 1 .063 6.3 3.50 to less than 4.00 6 .375 37.5 4.00 to less than 4.50 6 .375 37.5 4.50 to less than 5.00 2 .133 13.3 5.00 to less than 5.50 1 .06 62.28 a. & b. Turnovers Frequency Relative Frequency Percentage 0 4 .167 16.7 1 5 .208 20.8 2 7 .292 29.2 3 5 .208 20.8 4 3 .125 12.5 c. The total for games with two or more turnovers is 7+ 5 + 3 = 15. Mann - Introductory Statistics , Fifth Edition, Solutions Manual 17 d. 9 y c n 6 e u q e 3 r F 0 0 1 2 3 4 Turnovers 2.29 a. & b. Number of Children less than Frequency Relative Frequency Percentage 18 Years of Age 0 8 .267 26.7 1 10 .333 33.3 2 10 .333 33.3 3 2 .067 6.7 c. 10 + 2 = 12 families have 2 or 3 children under 18 years of age. d. 12 y c 9 n e u 6 q e r 3 F 0 0 1 2 3 Number of children 2.3060 65 y y c c 55 n n 40 e e u 45 u q q 20 e e r r 35 F F 0 25 0 1 2 3 4 0 1 2 3 4 Number of Tickets Number of Tickets The truncated graph exaggerates the difference in the number of students with different numbers of tickets. 2.31 18 22 y 19 c n e 16 u q 13 e rF 10 0-6 6-12 12-18 18-24 24-30 Time The graph with the truncated frequency axis exaggerates the differences in the frequencies of the various classes.2.32 The cumulative frequency distribution gives the total number of values that fall below the upper boundary of each class. The cumulative relative frequencies are obtained by dividing the cumulative frequencies by the total number of observations in the data. The cumulative percentages are obtained by multiplying the cumulative relative frequencies by 100.2.33 An ogive is drawn for a cumulative frequency distribution, a cumulative relative frequency distribution, or a cumulative percentage distribution. An ogive can be used to find the approximate cumulative frequency (cumulative relative frequency or cumulative percentage) for any class interval.2.34 a. & b. Number of Checks Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 0 to 99 39 .39 39 0 to 199 60 .60 60 0 to 299 78 .78 78 0 to 399 93 .93 93 0 to 499 100 1.00 100 c. 60% of the students wrote 199 or fewer checks in 2002. d. e. About 69% of the students wrote 250 or fewer checks in 2002.2.35 a. & b. Age (years) Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency Mann - Introductory Statistics , Fifth Edition, Solutions Manual 1918 to 30 12 .24 24 18 to 43 31 .62 62 18 to 56 45 .90 90 18 to 69 50 1.00 100 c. 100 – 62 = 38% of the employees are 44 years of age or older. d. e. About 52% of the employees are 40 years of age or younger.2.36 Team Revenue Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 125 to 130 2 .133 13.3 125 to 136 7 .467 46.7 125 to 142 11 .733 73.3 125 to 148 13 .867 86.7 125 to 154 13 .867 86.7 125 to 160 15 1.000 100.02.37 Number of Computer Cumulative Frequency Cumulative Relative Cumulative Percentage Terminals Manufactured Frequency 21 – 23 7 .233 23.3 21 – 26 11 .367 36.7 21 – 29 20 .667 66.7 21 – 32 24 .800 80.0 21 – 35 30 1.00 100.02.38 Keyboards Assembled Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 41 – 44 5 .20 20 41 – 48 13 .52 52 41 – 52 21 .84 84 41 – 56 25 1.00 1002.39 Telephone Bill (dollars) Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 20 15 to less than 19 4 .16 16 15 to less than 23 15 .60 60 15 to less than 27 22 .88 88 15 to less than 31 24 .96 96 15 to less than 35 25 1.00 1002.40 Visit to Doctor (dollars) Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 50 to less than 60 9 .36 36 50 to less than 70 16 .64 64 50 to less than 80 23 .92 92 50 to less than 90 25 1.00 1002.41 Price of Beer (dollars) Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 6.30 – 6.90 4 .16 16 6.30 – 7.50 17 .68 68 6.30 – 8.10 23 .92 92 6.30 – 8.70 25 1.00 1002.42 Median Income Cumulative Frequency Cumulative Relative Cumulative Percentage (thousands of dollars) Frequency 30 to less than 35 4 .211 21.1 30 to less than 40 11 .579 57.9 30 to less than 45 13 .684 68.4 30 to less than 50 17 .899 89.5 30 to less than 55 19 1.000 100.0About 12 states in the table have median household income less than $43,000 for that year.2.43 ERA Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 3.00 to less than 3.50 1 .063 6.3 3.00 to less than 4.00 7 .438 43.8 3.00 to less than 4.50 13 .813 81.3 Mann - Introductory Statistics , Fifth Edition, Solutions Manual 213.00 to less than 5.00 15 .938 93.8 3.00 to less than 5.50 16 1.000 100.0Approximately 11 of the teams had an ERA of less than 4.20.2.44 To prepare a stem and leaf display for a data set, each value is divided into two parts – the first part is called the stem and the second part is called the leaf. The stems are written on the left side of a vertically drawn line and the leaves for each stem are written on the right side of the vertical line next to the corresponding stem. Exercise 2.46 is an example. 2.45 The advantage of a stem-and-leaf display over a frequency distribution is that by preparing a stem-and- leaf display we do not lose information on individual observations. From a stem-and-leaf display we can obtain the original data. However, we cannot obtain the original data from a frequency distribution table. Consider the following stem-and-leaf display.3 3 5 6 6 4 2 4 4 7 9 9 5 0 1 1 3 4 5 5The data that were used to make this stem-and-leaf display are: 33, 35, 36, 36, 42, 44, 44, 47, 49, 49, 50, 51, 51, 53, 54, 55, 55 2.46 43 46 50 51 54 55 63 64 66 67 67 67 68 69 72 72 73 75 76 76 79 80 87 88 892.47 218 245 256 329 367 383 397 404 427 433 471 523 537 551 563 581 592 622 636 647 655 678 689 810 8412.48 0 8 5 6 5 3 0 3 5 5 6 8 1 5 7 0 4 7 6 9 1 0 4 5 6 7 7 9 2 3 1 2 5 2 1 2 3 5 3 1 4 0 1 3 0 1 1 42.49 7 45 75 7 45 75 8 48 00 57 8 00 48 57 9 21 33 67 95 9 21 33 67 95 10 24 09 1 09 24 0 22 11 33 45 1 33 45 1 12 75 1 75 22.50 2 4 7 3 9 5 3 8 1 6 2 7 7 3 8 9 2 6 8 3 7 3 2 3 3 1 3 1 5 4 5 12 1 2 2 3 3 3 3 4 5 6 6 7 7 7 7 8 8 8 9 9 3 1 1 1 2 3 3 3 4 5 52.51 4 5 8 1 6 4 2 8 8 6 3 7 4 7 9 5 2 6 3 1 3 1 2 0 4 0 24 1 2 3 4 4 5 6 6 7 7 8 8 8 9 5 0 0 1 1 2 2 2 3 3 4 62.52 Cost of a visit to the doctor rounded to the nearest dollar: 52 89 55 70 75 72 80 64 57 56 78 59 72 64 71 53 58 59 60 72 63 68 60 62 65Stem-and-leaf display: Ranked stem-and-leaf display: 5 2 5 7 6 9 3 8 9 5 2 3 5 6 7 8 9 9 6 4 4 0 3 8 0 2 5 6 0 0 2 3 4 4 5 8 7 0 5 2 8 2 1 2 7 0 1 2 2 2 5 8 8 9 0 8 0 92.53 0 5 7 0 5 7 1 0 1 7 5 9 1 0 1 5 7 9 2 3 6 6 9 1 2 2 1 2 3 6 6 9 3 3 9 2 3 2 3 9 4 8 3 4 3 8 5 0 5 0 6 5 6 52.54 a. 0 6 5 8 9 1 0 2 5 3 8 2 0 5 4 0 4 6 2 8 3 7 0 3 6 6 2 8 4 0 5 7 2 0 2 8 0 4 6 9 6 2 0 b. 0-2 6 5 8 9 * 0 2 5 3 8 * 0 5 4 0 4 6 2 8 3-5 7 0 3 6 6 2 8 * 0 5 * * 6-9 * * 2 0 2 * 0 4 6 * 6 2 02.55 a. 1 58 2 10 20 45 65 68 70 Mann - Introductory Statistics , Fifth Edition, Solutions Manual 233 20 45 50 68 90 4 30 38 57 60 75 87 90 5 05 28 30 38 40 60 65 70 6 17 35 38 7 02 05 06 20 21 b. 1-3 58 * 10 20 45 65 68 70 * 20 45 50 68 90 4-5 30 38 57 60 75 87 90 * 05 28 30 38 40 60 65 70 6-7 17 35 38 * 02 05 06 20 212.56 a. & b.Political Party Frequency Relative Frequency Percentage D 9 .300 30.0 DR 4 .133 13.3 F 2 .067 6.7 R 11 .367 36.7 W 4 .133 13.3 c. y 0.4 c n e W u 0.3 q 1 3 . 3 % D e r 0.2 3 0 . 0 % F e v 0.1 R i t 3 6 . 7 % D R a l 0 1 3 . 3 % e F R D DR F R W 6 . 7 % Political Party d. 13.3% of these presidents were Whigs.2.57 a. & b.Favorite Sport Frequency Relative Frequency Percentage CF 5 .139 13.9 CB 3 .083 8.3 MLB 9 .250 25.0 NFL 13 .361 36.1 NBA 6 .167 16.7 c.15 CF NBA 13.9% 12 16.7% CB y c 8.3% n 9 e u 6 q e r 3 NFL MLB F 0 36.1% 25.0% CF CB MLB NFL NBA Favorite Sport d. 25% of these people prefer Major League Baseball. 24 2.58 a. & b. TV sets owned Frequency Relative Frequency Percentage 0 1 .025 2.5 1 14 .350 35.0 2 14 .350 35.0 3 8 .200 20.0 4 3 .075 7.5 c. 15 12 y c n 9 e u 6 q e r 3 F 0 0 1 2 3 4 TVs owned d. 62.5% of the households own 2 or more television sets.2.59 a. & b. Correct Names Frequency Relative Frequency Percentage 0 1 .042 4.2 1 3 .125 12.5 2 4 .167 16.7 3 6 .250 25.0 4 4 .167 16.7 5 6 .250 25.0 c. 4.2 + 12.5 = 16.7% of the students name less than two representatives correctly. d. y 0.3 c n e u 0.2 q e r F e 0.1 v i t a l 0 eR 0 1 2 3 4 5 Correct Names 2.60 a. & b. Amount Spent on Video Frequency Relative Frequency Percentage Rentals (in dollars) 1 – 200 20 .667 66.7 201 – 400 4 .133 13.3 Mann - Introductory Statistics , Fifth Edition, Solutions Manual 25401 – 600 3 .100 10.0 601 – 800 2 .067 6.7 801 – 1000 1 .033 3.3 c. 10.0 + 6.7 + 3.3 = 20.0% of the households in this sample spent more than $400 on video rentals.2.61 a. & b. Number of Orders Frequency Relative Frequency Percentage 23 – 29 4 .133 13.3 30 – 36 9 .300 30.0 37 – 43 6 .200 20.0 44 – 50 8 .267 26.7 51 – 57 3 .100 10.0 c. For 20.0 + 26.7 + 10.0 = 56.7% of the hours in this sample, the number of orders was more than 36.2.62 a. & b. Concession (dollars) Frequency Relative Frequency Percentage 0 to less than 6 9 .300 30.0 6 to less than 12 10 .333 33.3 12 to less than 18 5 .167 16.7 18 to less than 24 4 .133 13.3 24 to less than 30 2 .067 6.7 c. 10 y c n e 5 u q e r 0 F 0-6 6-12 12-18 18-24 24-30 Concession Expenditures2.63 a. & b. Car Repair Costs Frequency Relative Frequency Percentage (dollars) 1 – 1400 11 .367 36.7 1401 – 2800 10 .333 33.3 2801 – 4200 3 .100 10.0 4201 – 5600 2 .067 6.7 5601 – 7000 4 .133 13.3 c. 26 y 0.4 c n e 0.3 u q e 0.2 r F e 0.1 v i t a 0 l eR 1- 1401- 2801- 4201- 5601- 1400 2800 4200 5600 7000 Car Repair Costs d. The class boundaries of the fourth class are $4200.50 and 5600.50. The width of this class is $1400.2.64 a. & b. Amount Spent on Video Cumulative Frequency Cumulative Relative Cumulative Percentage Rentals (in dollars) Frequency 1 – 200 20 .667 66.7 1 – 400 24 .800 80.0 1 – 600 27 .900 90.0 1 – 800 29 .967 96.7 1 – 1000 30 1.000 100.02.65 Number of Orders Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 23 – 29 4 .133 13.3 23 – 36 13 .433 43.3 23 – 43 19 .633 63.3 23 – 50 27 .900 90.0 23 – 57 30 1.000 100.02.66 Concession(dollars) Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 0 to less than 6 9 .300 30.0 0 to less than 12 19 .633 63.3 0 to less than 18 24 .800 80.0 0 to less than 24 28 .933 93.3 0 to less than 30 30 1.000 100.02.67 Car Repair Costs Cumulative Frequency Cumulative Relative Cumulative Percentage (dollars) Frequency 1 – 1400 11 .367 36.7 1 – 2800 21 .700 70.0 1 – 4200 24 .800 80.0 1 – 5600 26 .867 86.7 1 – 7000 30 1.000 100.02.68 0 24 06 40 90 55 90 70 88 83 22 70 15 1 00 00 55 80 00 27 60 11 Mann - Introductory Statistics , Fifth Edition, Solutions Manual 272 05 40 3 10 50 4 05 05 5 95 6 22 7 60 8 082.69 2 8 4 7 7 3 4 1 8 5 2 9 3 7 0 8 4 6 0 4 4 1 7 6 1 9 5 6 7 5 2 3 7 02.70 Let the six categories listed in the table be denoted by S, ML, M, O, V, and F respectively. ) s d ) 1100 n s 1200 a d900 s n e u a c 600 i s o 800 r h u tP 300 ( o h0 e t ( c i 500 S ML M O V F r P Town S ML M O V F TownThe truncated graph exaggerates the differences in the sale prices of houses in these towns.2.71 Let the seven categories listed in the table be denoted by LA, TX, MI, OK, AL, GA, and SC respectively. e e t 800 t 800 a a R R n n 700 o o i i t 400 t a a r r 600 e e c c r r a a c 0 c 500 n n I LA TX MI OK AL GA SC I LA TX MI OK AL GA SC State StateThe truncated graph exaggerates the differences in the incarceration rates for the seven states.2.72 a. Age Frequency Relative Frequency 18 to less than 20 7 .060 20 to less than 25 12 .103 25 to less than 30 18 .154 30 to less than 40 14 .120 40 to less than 50 15 .128 50 to less than 60 16 .137 28 60 and over 35 .299 b. & c. This histogram is misleading because the class widths differ. If you were to change the frequency distribution to reflect equal class widths, the resulting histogram would give a clearer picture.2.73 The greater relative frequency of accidents in the older age group does not imply that they are more accident-prone than the younger group. The older group may drive more miles during a week than the younger group. 2.74 a. Sturge’s formula: c = 1+3.3 log n where c = number of classes and n = number of observations in data set. c = 1 + 3.3 log (135)  8 b. Class width = (53-20)/8 = 4.125 which we round up to a class width of 5.2.75 a.1 7 8 8 8 8 9 9 9 9 9 9 9 9 9 2 0 0 0 0 0 0 0 1 1 1 2 2 3 6 8 3 0 8 4 3 5 1 6 2 4 b. Enrolled Students Frequency 15 to less than 25 27 25 to less than 35 3 35 to less than 45 2 45 to less than 55 1 45 to less than 65 2The data is very skewed to the left. 77% shows up in the first class. Mann - Introductory Statistics , Fifth Edition, Solutions Manual 29 c. Enrolled Students Frequency 17 to less than 19 5 19 to less than 21 16 21 to less than 23 5 23 to less than 25 1 25 to less than 35 3 35 to less than 45 2 45 and over 32.76 a. Answers will vary. b. i. 9 9 10 2 8 8 11 0 4 5 5 6 9 12 3 3 3 5 8 8 13 2 3 8 14 6 7 7 8 15 5 9 16 1 2 4 8 17 4 4 5 9 9 9 18 0 2 3 9 19 3 3 5 20 2 4 ii. The display shows a bimodal distribution, due to the presence of both females and males in the sample. The males tend to be heavier, so their weights are concentrated in the larger values, while the females’ weights are found primarily in the smaller values.Self-Review Test for Chapter Two1. An ungrouped data set contains information on each member of a sample or population individually. The following data, which give the grade point averages of eight students, is an example of ungrouped data. 3.45 2.98 3.81 2.04 3.67 3.01 1.88 2.59Data presented in the form of a frequency table are called grouped data. The following table gives the frequency distribution table for the grade point averages of 100 students. This is an example of grouped data. 30 Grade Point Average Frequency 1.5 to less than 2.0 5 2.0 to less than 2.5 13 2.5 to less than 3.0 34 3.0 to less than 3.5 31 3.5 to less than 4.0 172. a. 5 b. 7 c. 17 d. 6.5 e. 13 f. 90 g. .303. A histogram that is identical on both sides of its central point is called a symmetric histogram. A histogram that is skewed to the right has a longer tail on the right side, and a histogram that is skewed to the left has a longer tail on the left side. The following three histograms present these three cases.4. a. & b.Category Frequency Relative Frequency Percentage B 8 .40 40 F 4 .20 20 M 7 .35 35 S 1 .05 5 c. 35% of the children live with their mothers only. Mann - Introductory Statistics , Fifth Edition, Solutions Manual 31 d.S 5%B M 40% 35%F 20%5. a. & b.Number of Years Frequency Relative Frequency Percentage 1 – 4 5 .208 20.8 5 – 8 6 .250 25.0 9 – 12 6 .250 25.0 13 – 16 5 .208 20.8 17 – 20 2 .083 8.3 c. 20.8 + 25.0 = 45.8% of the employees have been with their current employer for 8 or fewer years. d. 8 y c 6 n e 4 u q e 2 rF 0 1-4 5-8 9-12 13-16 17-20 Number of Years6. a. & b Number of Years Cumulative Frequency Cumulative Relative Cumulative Percentage Frequency 1 – 4 5 .208 20.8 1 – 8 11 .458 45.8 1 – 12 17 .708 70.8 1 – 16 22 .917 91.7 1 – 20 24 1.000 100.0 32 About 61 % of the employees have been with their current employer for 11 or fewer years. 7. 0 4 6 7 8 1 0 2 2 3 4 4 5 6 6 6 7 8 9 2 0 1 2 2 5 9 3 28. 30 33 37 42 44 46 47 49 51 53 53 56 60 67 67 71 79

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