Chapter Two Section 2.1 – 2.2 2.1
Data in their original form are often too large and unmanageable. It is easier to make sense of grouped data than ungrouped data and easier to make decisions and draw conclusions using grouped 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. Example 2−2 in the text is an example which shows how relative frequencies and percentages are calculated.
2.3
a. & b. Category A B C
Frequency 8 8 14
Relative Frequency 8/30 = .267 8/30 = .267 14/30 = .467
Percentage 26.7 26.7 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.
Frequency
16 12 8 4 0 A
B
C
Category
2.4
a. & b. Category Y N D
Frequency 23 13 4
Relative Frequency 23/40 = .575 13/40 = .325 4/40 = .100
c. 57.5% of the elements belong to category Y.
9
Percentage 57.5 32.5 10.0
10
Chapter Two
d. 32.5 + 10.0 = 42.5% of the elements belong to categories N or D. e. D 10.0%
N 32.5%
2.5
Y 57.5%
a. & b. Category F SO J SE
Frequency 12 12 15 11
Relative Frequency 12/50 = .24 12/50 = .24 15/50 = .30 11/50 = .22
Percentage 24 24 30 22
c. 30 + 22 = 52% of the students are juniors or seniors. d.
Frequency
16 12 8 4 0 F
SO
J
SE
Category
2.6
a. & b. Category T R A P M
Frequency 4 10 7 8 1
Relative Frequency 4/30 = .133 10/30 = .333 7/30 = .233 8/30 = .267 1/30 = .033
Percentage 13.3 33.3 23.3 26.7 3.3
c. 33.3 + 23.3 = 56.6% of the adults ranked refrigerators or air conditioning as the convenience that they would find most difficult to do without.
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
Relative Frequency
d. 0.40 0.30 0.20 0.10 0.00 T
R
A
P
M
Category
2.7
a. & b. Category H C O
Frequency 10 6 4
Relative Frequency 10/20 = .50 6/20 = .30 4/20 = .20
Percentage 50 30 20
c. 50% of the dieticians gave Health as the major reason for people to lose weight. d.
O 20.0%
C 30.0%
H 50.0%
2.8
a. & b. Category C CK CC D O
Frequency 4 5 4 2 1
c. CC 25.0%
D 12.5%
C 25.0% CK 31.3%
O 6.3%
Relative Frequency 4/16 = .250 5/16 = .313 4/16 = .250 2/16 = .125 1/16 = .063
Percentage 25.0 31.3 25.0 12.5 6.3
11
12
Chapter Two
2.9
Let the seven categories listed in the table be denoted by PF, B, CF, AR, MPB, H, and MCB respectively, and let O represent "Other".
H 5.0%
MCB 5.0% O 17.0%
MPB 6.0% AR 8.0%
PF 31.0%
CF 12.0% B 16.0%
Let the five categories listed in the table be denoted by NG, HG, SC, CG, and PG respectively.
Percentage
2.10
35 30 25 20 15 10 5 0 NG
HG
SC
CG
PG
Category
Section 2.3 – 2.4 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. Example 2-4 is an example that illustrates 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. Example 2−4 is an example of the writing classes using limits method. A data set that contains fractional values is grouped by using the less than method. Example 2−5 is an example of the less than method. Single-valued classes are used to group a data set that contains only a few distinct (integer) values. Example 2−6 is an example of the single-valued classes method.
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
2.14
a. & c. Class Boundaries -0.5 to less than 3.5 3.5 to less than 7.5 7.5 to less than 11.5 11.5 to less than 15.5 15.5 to less than 19.5
Class Midpoint 1.5 5.5 9.5 13.5 17.5
Relative Frequency .225 .325 .275 .138 .038
Percentage 22.5 32.5 27.5 13.8 3.8
b. Yes, each class has a width of 4. d. 27.5 +13.8 + 3.8 = 45.1% of the adults possess 8 or more credit cards. 2.15
a. & c. Class Boundaries 17.5 to less than 30.5 30.5 to less than 43.5 43.5 to less than 56.5 56.5 to less than 69.5
Class Midpoint 24 37 50 63
Relative Frequency .24 .38 .28 .10
Percentage 24 38 28 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 1 to 200 201 to 400 401 to 600 601 to 800 801 to 1000 1001 to 1200
2.17
Class Midpoints 100.5 300.5 500.5 700.5 900.5 1100.5
Class Boundaries .5 to less than 25.5 25.5 to less than 50.5 50.5 to less than 75.5 75.5 to less than 100.5 100.5 to less than 125.5 125.5 to less than 150.5
Class Width 25 25 25 25 25 25
a., b., & c. Class Limits 1 to 25 26 to 50 51 to 75 76 to 100 101 to 125 126 to 150
2.18
Class Boundaries .5 to less than 200.5 200.5 to less than 400.5 400.5 to less than 600.5 600.5 to less than 800.5 800.5 to less than 1000.5 1000.5 to less than 1200.5
Class Midpoint 13 38 63 88 113 138
a. & b. Median Income 36,000 to 40,999 41,000 to 45,999 46,000 to 50,999 51,000 to 55,999 56,000 to 60,999 61,000 to 65,999 c.
Frequency 6 9 18 6 7 5
The data are skewed slightly to the right.
Relative Frequency .118 .176 .353 .118 .137 .098
Percentage 11.8 17.6 35.3 11.8 13.7 9.8
13
14
Chapter Two
d.
11.8 + 17.6 + 35.3 + 11.8 = 76.5% of these states had a median household income of less than $56.000.
2.19
a. & b. Number of Computer Monitors Manufactured 21 to 23 24 to 26 27 to 29 30 to 32 33 to 35
Frequency
Relative Frequency
Percentage
7 4 9 4 6
.233 .133 .300 .133 .200
23.3 13.3 30.0 13.3 20.0
c. 35 Percentage
30 25 20 15 10 5 0 21-23 24-26 27-29 30-32 33-35 Number of Computer Monitors
d.
a. & b. Keyboards Assembled 41 to 44 45 to 48 49 to 52 53 to 56
Frequency 5 8 8 4
Relative Frequency .20 .32 .32 .16
c. & d. Relative Frequency
2.20
For 30% of the days, the number of computer monitors manufactured is in the interval 27 to 29.
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 41-44
45-48
49-52
53-56
Number of Keyboards Assembled
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
2.21
2.22
a. & b. Charitable Contributions (millions of dollars) 75 to less than 125 125 to less than 175 175 to less than 225 225 to less than 275 275 to less than 325 325 to less than 375 a. & b.
Frequency
Relative Frequency
Percentage
10 5 0 4 1 2
.455 .227 .000 .182 .045 .091
45.5 22.7 0.0 18.2 4.5 9.1
15
The minimum colon and rectum cancer rate for women is 35.2, and the maximum rate is 51.5. The following table groups these data into six classes of equal width (3) with a starting point of 35.0.
Colon & Rectum Cancer Rates (Females) 35.0 to less than 38.0 38.0 to less than 41.0 41.0 to less than 44.0 44.0 to less than 47.0 47.0 to less than 50.0 50.0 to less than 53.0 2.23
a. & b.
Frequency
Relative Frequency
Percentage
3 2 9 2 4 2
.136 .091 .409 .091 .182 .091
13.6 9.1 40.9 9.1 18.2 9.1
The minimum colon and rectum cancer rate for men is 47.5, and the maximum rate is 71.3. The following table groups these data into six classes of equal width (4) with a starting point of 47.5.
Colon & Rectum Cancer Rates (Males) 47.5 to less than 51.5 51.5 to less than 55.5 55.5 to less than 59.5 59.5 to less than 63.5 63.5 to less than 67.5 67.5 to less than 71.5 2.24
a. & b.
Frequency
Relative Frequency
Percentage
2 5 3 4 5 3
.091 .227 .136 .182 .227 .136
9.1 22.7 13.6 18.2 22.7 13.6
The minimum lung and bronchus cancer rate for women is 20.9, and the maximum rate is 71.4. The following table groups these data into six classes of equal width (9) with a starting point of 20.5.
Lung & Bronchus Cancer Rates (Females) 20.5 to less than 29.5 29.5 to less than 38.5 38.5 to less than 47.5 47.5 to less than 56.5 56.5 to less than 65.5 65.5 to less than 74.5
Frequency
Relative Frequency
Percentage
1 2 5 5 8 1
.045 .091 .227 .227 .364 .045
4.5 9.1 22.7 22.7 36.4 4.5
16
Chapter Two
Relative Frequency
c. 0.4 0.3 0.2 0.1 0.0 20.5-29.5
29.5-38.5
38.5-47.5
47.5-56.5
56.5-65.5
65.5-74.5
Lung & Bronchus Cancer Rates for Women
2.25
a. & b.
The minimum lung and bronchus cancer rate for men is 40.3, and the maximum rate is 113.7. The following table groups these data into six classes of equal width (13) with a starting point of 40.0.
Lung & Bronchus Cancer Rates (Males) 40.0 to less than 53.0 53.0 to less than 66.0 66.0 to less than 79.0 79.0 to less than 92.0 92.0 to less than 105.0 105.0 to less than 118.0
Frequency
Relative Frequency
Percentage
1 3 5 9 1 3
.045 .136 .227 .409 .045 .136
4.5 13.6 22.7 40.9 4.5 13.6
Relative Frequency
c. 0.5 0.4 0.3 0.2 0.1 0.0 40.0-53.0
53.0-66.0
66.0-79.0
79.0-92.0 92.0-105.0 105.0-118.0
Lung & Bronchus Cancer Rates for Men
2.26
a. & b.
The minimum non-Hodgkin lymphoma cancer rate for women is 13.2, and the maximum rate is 18.3. The following table groups these data into four classes of equal width (1.5) with a starting point of 13.0.
Non-Hodgkin Lymphoma Cancer Rates (Females) 13.0 to less than 14.5 14.5 to less than 16.0 16.0 to less than 17.5 17.5 to less than 19.0
Frequency
Relative Frequency
Percentage
3 9 7 3
.136 .409 .318 .136
13.6 40.9 31.8 13.6
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
c. Relative Frequency
0.5 0.4 0.3 0.2 0.1 0.0 13.0-14.5
14.5-16.0
16.0-17.5
17.5-19.0
Non-Hodgkin Lymphoma Cancer Rates for Women 2.27
a. & b Points Scored 12.0 to less than 15.5 15.5 to less than 19.0 19.0 to less than 22.5 22.5 to less than 26.0 26.0 to less than 29.5
2.28
Frequency 2 3 3 7 1
Relative Frequency .125 .188 .188 .438 .063
Percentage 12.5 18.8 18.8 43.8 6.3
Frequency 4 5 7 5 3
Relative Frequency .167 .208 .292 .208 .125
Percentage 16.7 20.8 29.2 20.8 12.5
a. & b. Turnovers 0 1 2 3 4
c. 7 + 5 + 3 = 15 games had two or more turnovers. d.
Frequency
9 6 3 0 0
1
2
3
4
Turnovers
2.29
a. & b. Number of Errors in Credit Reports 0 1 2 3 4 5
Frequency
Relative Frequency
Percentage
7 7 5 3 2 1
.28 .28 .20 .12 .08 .04
28 28 20 12 8 4
17
18
Chapter Two
c.
5 + 3 + 2 + 1 = 11 credit reports have two or more errors.
d.
Frequency
9 6 3 0 0
1
2
3
4
5
Number of Errors
2.30
65 Frequency
Frequency
60 40 20
55 45 35 25
0 0
1
2
3
0
4
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.
25
22
20
19
Frequency
Frequency
2.31
15 10 5 0
16 13 10
0-6
6-12 12-18 18-24 24-30 Time
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 time intervals.
Section 2.5 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 set. The cumulative percentages are obtained by multiplying the cumulative relative frequencies by 100.
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
2.33
19
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 Credit Cards 0 to 3 0 to 7 0 to 11 0 to 15 0 to 19
Cumulative Frequency 18 44 66 77 80
Cumulative Relative Frequency .225 .550 .825 .963 1.000
Cumulative Percentage 22.5 55.0 82.5 96.3 100.0
c. 55% of the adults possessed seven or fewer credit cards.
Cumulative Percentage
d. 120 100 80 60 40 20 0 -0.5
3.5
7.5
11.5 15.5 19.5
Number of Credit Cards
e. Approximately 75% of the adults possessed 10 or fewer credit cards, as indicated on the ogive in part d. 2.35
a. & b. Age (years)
Cumulative Frequency
18 to 30 18 to 43 18 to 56 18 to 69
12 31 45 50
Cumulative Relative Frequency .24 .62 .90 1.00
c. 100 – 62 = 38% of the employees are 44 years of age or older.
Cumulative Percentage 24 62 90 100
20
Chapter Two
Cumulative Percentage
d. 120 100 80 60 40 20 0
17.5
30.5
43.5
56.5
69.5
Age
e. Approximately 52% of the employees are 40 years of age or younger as indicated on the ogive in part d. 2.36 Median Income
Cumulative Frequency
36,000 to 40.999 36,000 to 45,999 36,000 to 50,999 36,000 to 55,999 36,000 to 60,999 36,000 to 65,999
6 15 33 39 46 51
Cumulative Relative Frequency .118 .294 .647 .765 .902 1.000
Cumulative Percentage
Cumulative Relative Frequency .233 .367 .667 .800 1.000
Cumulative Percentage
Cumulative Relative Frequency .20 .52 .84 1.000
Cumulative Percentage
11.8 29.4 64.7 76.5 90.2 100.0
2.37 Number of Computer Monitors Manufactured 21 to 23 21 to 26 21 to 29 21 to 32 21 to 35
Cumulative Frequency 7 11 20 24 30
23.3 36.7 66.7 80.0 100.0
2.38 Keyboards Assembled 41 to 44 41 to 48 41 to 52 41 to 56
Cumulative Frequency 5 13 21 25
20 52 84 100
2.39 Colon & Rectum Cancer Rates (Males) 47.5 to less than 51.5 47.5 to less than 55.5 47.5 to less than 59.5 47.5 to less than 63.5 47.5 to less than 67.5 47.5 to less than 71.5
Cumulative Frequency 2 7 10 14 19 22
Cumulative Relative Frequency .091 .318 .455 .636 .864 1.000
Cumulative Percentage 9.1 31.8 45.5 63.6 86.4 100.0
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
2.40 Lung & Bronchus Cancer Rates (Males) 40.0 to less than 53.0 40.0 to less than 66.0 40.0 to less than 79.0 40.0 to less than 92.0 40.0 to less than 105.0 40.0 to less than 118.0
Cumulative Frequency
Cumulative Relative Frequency .045 .182 .409 .818 .864 1.000
Cumulative Percentage 4.5 18.2 40.9 81.8 86.4 100.0
Non-Hodgkin Lymphoma Cancer Rates (Females) 13.0 to less than 14.5 13.0 to less than 16.0 13.0 to less than 17.5 13.0 to less than 19.0
Cumulative Frequency
Cumulative Relative Frequency
Cumulative Percentage
3 12 19 22
.136 .545 .864 1.000
13.6 54.5 86.4 100.0
Charitable Contributions 75 to less than 125 75 to less than 175 75 to less than 225 75 to less than 275 75 to less than 325 75 to less than 375
Cumulative Frequency
Cumulative Relative Frequency .455 .682 .682 .864 .909 1.000
Cumulative Percentage 45.5 68.2 68.2 86.4 90.9 100.0
1 4 9 18 19 22
2.41
Cumulative Frequency
2.42
10 15 15 19 20 22
25 20 15 10 5 0 75 125 175 225 275 325 375 Charitable Contributions (in millions of dollars)
Approximately 15 individuals made charitable contributions of $200 million or less. 2.43 Points Scored
Cumulative Frequency
12.0 to less than 15.5 12.0 to less than 19.0 12.0 to less than 22.5 12.0 to less than 26.0 12.0 to less than 29.5
2 5 8 15 16
Cumulative Relative Frequency .125 .313 .500 .938 1.000
Cumulative Percentage 12.5 31.3 50.0 93.8 100.0
21
Chapter Two
Cumulative Frequency
22
20 15 10 5 0 12.0 15.5 19.0 22.5 26.0 29.6 ERA
Approximately 6 of the teams scored 20 or fewer points per game.
Section 2.6 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 vertical line and the leaves for each stem are written on the right side of the vertical line next to the corresponding stem. Example 2-9 is an example of a stem-and-leaf display.
2.45
The advantage of a stem-and-leaf display over a frequency distribution is that by preparing a stem-andleaf 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 stem-and-leaf display from Example 2−8: 5 6 7 8 9
2 5 5 0 6
0 9 9 7 3
7 1 1 1 5
8 2 6 2
4 6 3 2
9 4 8
7 7
1
2
The data that were used to make this stem-and-leaf display are: 52, 50, 57, 65, 69, 61, 68, 64, 75, 79, 71, 72, 76, 79, 77, 71, 72, 80, 87, 81, 86, 83, 84, 87, 96, 93, 95, 92, 92, 98 2.46
The data that were used to make this stem-and-leaf display are: 43, 46, 50, 51, 54, 55, 63, 64, 66, 67, 67, 67, 68, 69, 72, 72, 73, 75, 76, 76, 79, 80, 87, 88, 89
2.47
The data that were used to make this stem-and-leaf display are: 218, 245, 256, 329, 367, 383, 397, 404, 427, 433, 471, 523, 537, 551, 563, 581, 592, 622, 636, 647, 655, 678, 689, 810, 841
2.48 0 1 2 3
8 5 3 1
5 7 1 4
6 0 2 0
5 4 5 1
3 7
6
9
0 1 2 3
3 0 1 0
5 4 2 1
5 5 3 1
6 6 5 4
8 7
7
9
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
23
2.49 7 8 9 10 11 12
45 48 21 24 33 75
75 00 33 09 45
57 67
7 8 9 10 11 12
95
45 00 21 09 33 75
75 48 33 24 45
57 67
95
2.50 2 3
4 2
7 3
3 3
9 1
5 3
3 1
8 5
1 4
6 5
2 1
7
7
3
8
9
2
6
8
3
7
2 3
1 1
2 1
2 1
3 2
3 3
3 3
3 3
4 4
5 5
6 5
6
7
7
7
7
8
8
8
9
9
4 5
5 2
8 6
1 3
6 1
4 3
2 1
8 2
8 0
6 4
3 0
7 2
4
7
9
4 5
1 0
2 0
3 1
4 1
4 2
5 2
6 2
6 3
7 3
7 4
8 6
8
8
9
2.51
2.52
Average median income rounded to the nearest thousand: 41 61 48 39 56 59 64 54 50 46 49 48 40 39 47 66 58 50 58 36 65 43 49 42 45 48 42 49 49 55 50 59 57 41 52 49 Stem-and-leaf display: 3 9 9 6 4 1 8 6 8 7 9 5 6 9 4 0 0 1 6 1 4 3 6 6 5
8 8
0 0
Ranked stem-and-leaf display: 3 6 9 9 4 0 1 1 2 2 2 2 3 5 0 0 0 0 1 2 4 4 6 1 3 4 5 6 6
50 46 42
5 0 3 3 8 0 5
7 1 6 9 3
7 6 2
5 9
9 1
2
48 49 42
51 54 45
47 66 55
7 8
6 4
3 5
9 5
3 0
9 9
2 7
5 2
8
2
9
9
2
7
2
5
1
9
3 5
5 5
5 6
6 7
6 8
7 8
7 9
7 9
8
8
8
8
9
9
9
9
9
9
0 1 2 3 4 5 6
5 0 1 2 3 0 5
7 1 2 3 8
5 3 9
7 6
9 6
9
2.53 0 1 2 3 4 5 6
63 43 47
24
Chapter Two
2.54
a. 0 1 2 3 4 5 6 7 8 9
6 0 0 7 0
5 2 5 0 5
8 5 4 3
2 0 6
0 4 2
2 6 0
9 3 0 6
8 4 6
6 2
2 8
8
b. 0-2 3-5 6-9 2.55
6 7 *
5 0 *
8 3 2
9 6 0
* 6 2
0 2 *
2 8 0
5 * 4
3 0 6
8 5 *
* * 6
0 * 2
5
75
38 75 *
57 * 45
4
0
4
6
2
8
0
a. 2 3 4 5 6 7 8
58 20 30 05 10 02 45
68 45 38 30 17 05 90
68 57 38 20 06
60 40 35 20
70 50 38 21
90 60
65
70
28
65
87
b. 2-4 5-6 7-8
58 05 02
68 30 05
* 38 06
20 40 20
45 50 21
68 60 28
* 65 65
30 70 87
60 10 90
70 17
90 20
35
38
Section 2.7 2.56
In order to prepare a dotplot, first we draw a horizontal line with numbers that cover the given data set. Then we place a dot above the value on the number line that represents each measurement in the data set. Example 2-11 illustrates this procedure.
2.57
A stacked dotplot is used to compare two or more data sets by creating a dotplot for each data set with numbers lines for all data sets on the same scale. The data sets are placed on top of each other.
2.58
0
1
2
3
4
5
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
25
2.59
40
45
50
55
Keyboards Assembled
2.60
0
1
2
3
4
Turnovers
2.61
0
1
2
3
4
5
Credit Report Errors
2.62
0
3
6
9
12
15
ATM Use
There are two clusters in the data; most of the values lie in the cluster between zero and five, with only three data points between seven and nine. The value 15 appears to be an outlier. 2.63
0
2
4
6
8
10
Fast-food - Males
The data for males is clustered in two groups with the first group having values from zero to five, and the second having values from seven to 10.
26
Chapter Two
2.64
0
2
4
6
8
10
Fast-food - Females
The data for females is also clustered in two groups with the first group having values from zero to two, and the second having values from four to six; 10 appears to be an outlier. With these clusters in different areas, it appears that the female students ate at fast-food restaurants less often than did males during a seven-day period. 2.65
0
5
10
15 HBP
20
25
30
The data set contains two clusters – the first from zero to one and the second from three to seven. The value 27 is an outlier for number of times hit by pitch.
Supplementary Exercises a. & b. Political Party D DR F R W
Frequency 9 4 2 11 4
Relative Frequency .300 .133 .067 .367 .133
Percentage 30.0 13.3 6.7 36.7 13.3
c. Relative Frequency
2.66
0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
W 13.3% D 30.0%
D
DR
F
R
Political Party
e.
13.3% of these presidents were Whigs.
W
R 36.7%
DR 13.3% F 6.7%
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
2.67
a. & b. Response L N C K
Frequency 11 12 1 16
Relative Frequency .275 .300 .025 .400
Percentage 27.5 30.0 2.5 40.0
c.
Frequency
20 15 10
L 27.5%
K 40.0%
5 0 L
N
C
K
Response
C 2.5%
N 30.0%
d. 27.5% of these respondents said “too liberal”. a. & b. TV sets owned 0 1 2 3 4
Frequency 1 14 14 8 3
Relative Frequency .025 .350 .350 .200 .075
Percentage 2.5 35.0 35.0 20.0 7.5
c.
Frequency
2.68
16 14 12 10 8 6 4 2 0 0
1
2
3
4
TV Sets Owned
d. 35.0 + 20.0 + 7.5 = 62.5% of the households own two or more television sets.
27
28
Chapter Two
2.69
a. & b. Correct Names 0 1 2 3 4 5
Frequency 1 3 4 6 4 6
Relative Frequency .042 .125 .167 .250 .167 .250
Percentage 4.2 12.5 16.7 25.0 16.7 25.0
c. 4.2 + 12.5 = 16.7% of the students named fewer than two of the representatives correctly.
Relative Frequency
d. 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0
1
2
3
4
5
Correct Names
2.70
a. & b. Amount Spent on Video Rentals (in dollars) 1 – 200 201 – 400 401 – 600 601 – 800 801 – 1000
Frequency
Relative Frequency
Percentage
20 4 3 2 1
.667 .133 .100 .067 .033
66.7 13.3 10.0 6.7 3.3
c. 10.0 + 6.7 + 3.3 = 20.0% of the households in this sample spent more than $400 on video rentals. 2.71
a. & b. Number of Orders 23 – 29 30 – 36 37 – 43 44 – 50 51 – 57
Frequency 4 9 6 8 3
Relative Frequency .133 .300 .200 .267 .100
Percentage 13.3 30.0 20.0 26.7 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. 2.72
a. & b. Concession (dollars) 0 to less than 6 6 to less than 12 12 to less than 18 18 to less than 24 24 to less than 30
Frequency 9 10 5 4 2
Relative Frequency .300 .333 .167 .133 .067
Percentage 30.0 33.3 16.7 13.3 6.7
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
29
Frequency
c. 12 10 8 6 4 2 0 0-6
6 - 12
12 - 18 18 - 24 24 - 30
Concessions
2.73
a. & b. Car Repair Costs (dollars) 1 – 1400 1401 – 2800 2801 – 4200 4201 – 5600 5601 – 7000
Frequency
Relative Frequency
Percentage
11 10 3 2 4
.367 .333 .100 .067 .133
36.7 33.3 10.0 6.7 13.3
Relative Frequency
c. 0.4 0.3 0.2 0.1 0
Car Repair Costs
d. The class boundaries of the fourth class are $4200.50 and $5600.50. The width of this class is $1400. 2.74
a. & b. Amount Spent on Video Rentals (in dollars) 1 – 200 1 – 400 1 – 600 1 – 800 1 – 1000
Cumulative Frequency 20 24 27 29 30
Cumulative Relative Frequency .667 .800 .900 .967 1.000
Cumulative Percentage 66.7 80.0 90.0 96.7 100.0
30
Chapter Two
2.75 Number of Orders
Cumulative Frequency
23 – 29 23 – 36 23 – 43 23 – 50 23 – 57
4 13 19 27 30
Concession (dollars)
Cumulative Frequency
0 to less than 6 0 to less than 12 0 to less than 18 0 to less than 24 0 to less than 30
9 19 24 28 30
Car Repair Costs (dollars) 1 – 1400 1 – 2800 1 – 4200 1 – 5600 1 – 7000
Cumulative Frequency
Cumulative Relative Frequency .133 .433 .633 .900 1.000
Cumulative Percentage
Cumulative Relative Frequency .300 .633 .800 .933 1.000
Cumulative Percentage
Cumulative Relative Frequency .367 .700 .800 .867 1.000
Cumulative Percentage
13.3 43.3 63.3 90.0 100.0
2.76
30.0 63.3 80.0 93.3 100.0
2.77
11 21 24 26 30
36.7 70.0 80.0 86.7 100.0
2.78 0 1 2 3 4 5 6 7 8
24 00 05 10 05 95 22 60 08
2 3 4 5
8 4 4 2
06 00 40 50 05
40 55
90 80
55 00
90 27
70 60
88 11
83
22
6
0
70
15
2.79 7 8 7 7
7 5 6 0
2 1
9 9
3 5
7 6
0 7
8
4
Let the six universities listed in the table be denoted by H, Y, S, P, U, and M respectively. 40
Endlowments (billions)
Endowments (billions)
2.80
4 1 1 3
30 20 10 0 H
Y
S
P
University
U
M
39 29 19 9 H
Y
S
P
U
University
The truncated graph exaggerates the differences in the endowments for the six universities.
M
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
Average Price/Gallon
4.00 3.00 2.00 1.00 0.00 99 00 01 02 03 04 05 06 07 08 Year
Average Price/Gallon
2.81 4.00 3.00 2.00 1.00 99 00 01 02 03 04 05 06 07 08 Year
The truncated graph exaggerates the differences in average price per gallon for the ten year period. 2.82
0
10
20
30
40
50
60
70
Time to Commute
2.83
20
30
40
50
60
Number of Orders
2.84
0
1
2
3
4
5
Correct Names 2.85
0
1
2 Number of Visitors
3
4
31
32
Chapter Two
2.86
a. Age 18 to less than 20 20 to less than 25 25 to less than 30 30 to less than 40 40 to less than 50 50 to less than 60 60 and over
Frequency 7 12 18 14 15 16 35
Relative Frequency .060 .103 .154 .120 .128 .137 .299
Relative Frequency
.300 .250 .200 .150 .100 .050
60 and over
50 to < 60
40 to < 50
30 to < 40
25 to < 30
20 to < 25
18 to < 20
.000
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. 2.87
The greater relative frequency of accidents in the older age group does not imply that they are more accident-prone than the younger group. For instance, the older group may drive more miles during a week than the younger group.
2.88
a. Using Sturge’s formula: c = 1 + 3.3 log n = 1 + 3.3 log 135 = 1 + 3.3(2.13033377) = 1 + 7.03 = 8.03 ≈ 8 b. Approximate class width = Use a class width of 5.
2.89
a. Answers will vary.
Largest value − smallest value 53 − 20 = = 4.125 . Number of classes 8
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
33
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.
2.90
a. Top Histogram – Endpoints: – 0.5, 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, 10.5; Width = 1 Bottom Histogram – Endpoints: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; Width = 1 b. There is one observation between the left endpoint of the interval and 8. This can be seen by overlaying the histograms and determining the counts for each interval of .5 on the x axis starting at the far left. The following table displays these frequencies: Interval 0.0 to less than 0.5 0.5 to less than 1.0 1.0 to less than 1.5 1.5 to less than 2.0 2.0 to less than 2.5 2.5 to less than 3.0 3.0 to less than 3.5 3.5 to less than 4.0 4.0 to less than 4.5 4.5 to less than 5.0 5.0 to less than 5.5 5.5 to less than 6.0 6.0 to less than 6.5 6.5 to less than 7.0 7.0 to less than 7.5 7.5 to less than 8.0 8.0 to less than 8.5 8.5 to less than 9.0 9.0 to less than 9.5 9.5 to less than 10.0 10.0 to les than 10.5
Frequency 2 2 5 6 6 3 4 5 1 4 3 1 0 1 2 1 2 0 0 2 0
34
Chapter Two
c. The leftmost bar in the first histogram is misleading because it makes it appear as though there are values in the data set less than zero. 2.91
90 100 110 120 130 140 150 160 170 180 190 200 210 Weight
90 100 110 120 130 140 150 160 170 180 190 200 210 Weight - Males
90 100 110 120 130 140 150 160 170 180 190 200 210 Weight - Females
The distribution of all weights is bimodal. The distribution of weights for males is skewed to the left while the distribution for females is skewed to the right. You cannot distinguish between the lightest males and heaviest females in the dotplot of all weights as the distributions overlap in the area between 130 and 170 pounds. 2.92
a. Fewer than 50% of the patients are in their 50s since the angle for that classification is slightly less than 180°. b. More than 75% of the patients are in their 50s and 60s since the angle for the total of the two classifications is slightly more than 270°. c. The mean and standard deviation of the patients’ ages as well as the mean and standard deviation of the ages of the population of men would be helpful. Stacked dotplots comparing the patients’ ages to ages in the population would assist in making comparisons. It is likely that there are more men in their 50s and 60s than in their 70s and 80s, and men in these age groups may be more likely to seek medical care than the younger or older groups.
2.93
a. The top money winners on the men’s tour tend to make more money per tournament than those on the women’s tour. Earnings on the men’s tour begin at $2300, and more of the data points are toward the higher end of the scale. Earnings on the women’s tour begin at $800, and more of the data points are toward the lower end of the scale.
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
35
b. Typical earnings per tournament played for the women’s tour would be around $2500; typical earnings per tournament played for the men’s tour would be around $3650. c. The data do not appear to have similar spreads for the two tours. Earnings on the men’s tour begin at $2300, the largest grouping is between $2300 and $4800, and go up to $9500. Earnings on the women’s tour begin at $800, the largest grouping is between $1100 and $2600, and only go up to $7500. d. On the women’s tour, the $7500 earnings level appears to be an outlier; on the men’s tour, both the $8700 and $9500 earnings levels appear to be outliers.
2.94
a. Figure 2.25(a) is the empirical CDF for the men’s tour and 2.25(b) is for the women’s tour. 1) On 2.25(a), the percentage of earning between $800 and $2300 is 0; 2) On 2.25(b), 100% is reached at $7500; 3) On 2.25(a), the graph takes a large number of vertical steps between $3000 and $5000. b. The long steps at the top of the graph indicate bigger gaps between observations indicating a few observations that pull the tail of the distribution to the right. c. Approximate values for $3000 – 30% for the men’s tour and 62% for the women’s tour Approximate values for $4000 – 57% for the men’s tour and 76% for the women’s tour Approximate percentage between $3000 and $4000 – 27% for the men’s tour and 14% for the women’s tour
2.95
a. Answers may include 6.6 and 7.6 because those values have the largest clusters. The value of 8.6 is in the center of the data. b. There is one outlier in the data. The value 13.8 is an outlier as it is in the tail of the distribution with a large gap preceding it. c. The distribution is skewed right as the majority of the values are between 6.2 and 10.6 with more values further to the right than to the left. d. We cannot conclude that Georgia had the highest obesity rate nor Vermont the lowest as these data represent the change in the obesity rates, not the actual rates.
2.96
a. The Midwest has the least variability as the data are clustered together. The South has the most variability as the data are the most widely spread. b. The South tends to have the highest obesity rates as a large number of the data points are above 27.0. The West and Northeast tend to have the lowest obesity rates with most of the data points below 25 and only one each above 27. c. The West appears to have have an outlier at approximately 18.4. The Northeast also appears to have an outlier at approximately 27.2.
2.97
a. The ACC received more than 25% of the vote. The section of the pie chart representing the ACC is
36
Chapter Two
more than one-quarter of the whole pie. b. Southeastern and Big East c. a = Conference USA, b = Pac 10, c = Others, d = Southeastern, e = Big East, f = Big 12, g = Big Ten, h = ACC.
Self-Review Test 1.
An ungrouped data set contains information on each member of a sample or population individually. The first part of Example 2-1 in the text, listing the responses of each of the 30 employees, is an example of ungrouped data. Data presented in the form of a frequency table are called grouped data. Table 2.4 in the solution of Example 2-1 is an example of grouped data.
2. a. 5
b. 7
c. 17
d. 6.5
e. 13.5
f. 90
g. .30
3. 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. Figure 2.8 in the text provides graphs of symmetric histograms, Figure 2.9a displays a histogram skewed to the right, and Figure 2.9b displays a histogram that is skewed to the left. 4. a. & b. Category B F M S
Frequency 8 4 7 1
Relative Frequency .40 .20 .35 .05
Percentage 40 20 35 5
c. 35% of the children live with their mothers only. d.
S 5%
Frequency
10 8
B 40%
M 35%
6 4 2 0 B
F
M
Parents
S
F 20%
Introductory Statistics, Mann, Seventh Edition - Instructor’s Solutions Manual
37
5. a. & b. Number of False Alarms 1– 3 4– 6 7– 9 10 – 12 13 – 15
Frequency 5 6 6 4 3
Relative Frequency .208 .250 .250 .167 .125
Percentage 20.8 25.0 25.0 16.7 12.5
c. 20.8 + 25.0 + 25.0 = 70.8% of the weeks had 9 or fewer false alarms. d. 7 Frequency
6 5 4 3 2 1 0 1-3
4-6 7-9 10-12 Number of False Alarms
13-15
6. Number of False Alarms Cumulative Frequency
Cumulative Percentage
1– 3 1– 6 1– 9 1 – 12 1 – 15
Cumulative Relative Frequency .208 .458 .708 .875 1.000
5 11 17 21 24
120 100 80 60 40 20 0
0.5
3.5
6.5
9.5 12.5 15.5
Age
7. 0 1 2 3
4 0 0 2
6 2 1
7 2 2
8 3 2
4 5
4 9
5
6
6
6
7
8
9
Cumulative Percentage 20.8 45.8 70.8 87.5 100.0
38
Chapter Two
8.
30 33 37 42 44 46 47 49 51 53 53 56 60 67 67 71 79
9.
0
3
6
9
12
15
