Mann - Introductory Statistics, Fifth Edition, Solutions Manual

2.1 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.

2.4 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.

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.

2.6 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.

2.7 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.

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

2.9 Let the seven categories listed in the table be denoted by S, HC, R, O, E, P, and U respectively.

2.10 Let the four categories listed in the table be denoted by E, D, B, and O respectively.

2.11 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 / 8

A 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 / 9

Single 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.

Number of Children / Frequency
0 / 13
1 / 26
2 / 38
3 / 18
4 / 5

2.14 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 / 1100.5

2.17 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

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 Terminals Manufactured / Frequency / Relative Frequency / Percentage
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

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.

2.21 a. & b.

Median Income
(thousands of dollars) / Frequency / Relative Frequency / Percentage
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 / 10.5

2.22 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 / 1237

2.23 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 / 33

2.24 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

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

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 / 8

2.27 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 / 6

2.28 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.

2.29 a. & b.

Number of Children less than 18 Years of Age / Frequency / Relative Frequency / Percentage
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.

2.30

The truncated graph exaggerates the difference in the number of students with different numbers of tickets.

2.31

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 Frequency / Cumulative Percentage
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.