CHAPTER 3
DESCRIBING DATA: NUMERICAL MEASURES
1.
  5.4 found by 27/5
2.
  5.5 found by 33/6
3.
a.
b.
Mean = 7.0, found by 28/4
(5  7)  (9  7)  (4  7)  (10  7)  0
4.
a.
b.
4.2 found by 21/5
(1.3  4.2)  (7.0  4.2)  (3.6  4.2)  (4.1  4.2)  (5.0  4.2)  0
5.
14.58, found by 43.74/3
6.
$20.95, found by $125.68/6
7.
a.
b.
15.4, found by 154/10
Population parameter since it includes all the salespersons at Midtown Ford.
8.
a.
b.
23.9, found by 167/7
Population parameter since it includes all the calls during a seven-day period.
9.
a.
b.
$54.55, found by $1091/20
A sample statistic, assuming that the power company serves more than 20 customers.
10.
a.
b.
10.73, found by 161/15
Sample of RN’s
11.
Yes, $162,900 found by 30($5430).
12.
Veteran sales people likely are above average and new recruits are below average. So, the sales
goal is impractical and leads to more exits.
13.
$22.91, found by
14.
$1.50 found by ($40 + $35)/50
15.
$17.75, found by [50($8) + 50($15) + 100($24)]/200 or $3550/200
16.
$143.75, found by ($1000 +$750 + $4000)/40
17.
a.
b.
c.
18.
Mean = 32.57; Median = 33; Mode = 15
300($20)  400($25)  400($23)
300  400  400

$25, 200
1100
no mode
The given value would be the mode
3 and 4, bimodal
27
Chapter 3
19.
Mean = 3.25; Median = 5; Mode = 5
20.
Mean = 11.1; Median = 10.5; Mode = 8
21.
a.
b.
22.
Median = 9.2 Modes are 8.2, 8.5, and 10.3
23.
Mean = 58.82; Median = 58; Mode = 58; All three measures are nearly identical.
24.
Mean = 94.5; Median = 99.5; Applicants are no better than regular people
25.
a. 7.53, found by 90.4/12
b. 7.45, found by (7.6 + 7.3)/2 and there are several modes
c. 8.5, found by 34/4 and 8.7, it is somewhat higher.
26.
Treat Wind Direction as nominal, Temperature as interval, and Pavement as ordinal data. That
would lead to a mod (Southwest) for the first column, a mean (91) for the second column, and a
median (Trace) for a third.
27.
12.8 found by
28.
6.0 percent increase, found by
29.
12.28 percent increase, found by
30.
20.4% found by
31.
2.46% found by
32.
5.11% found by
33.
34.
Median = 2.9
Mode = 2.9
5
(1.08)(1.12)(1.14)(1.26)(1.05)  1.128
8
8
(1.02)(1.08)(1.06)(1.04)(1.10)(1.06)(1.08)(1.04)  1.060
5
(1.094)(1.138)(1.117)(1.119)(1.147)  1.1228
66, 290, 000
1
14,968, 000
10
188.9
1
148.2
2.57
1
0.605
70
10.76% found by 5
1
42
29
11,354
1
4975
27,516
1
6.95% for private colleges found by 12
12, 284
7.12% for public colleges found by
12
The rates of increase are about the same with private colleges slightly higher.
Chapter 3
28
35.
a.
b.
c.
d.
7, found by 10 – 3
6, found by 30/5
2.4, found by 12/5
The difference between the highest number sold (10) and the smallest number sold (3) is
7. On the average the number of HDTV’s sold deviates by 2.4 from the mean of 6.
36.
a.
b.
c.
d.
24, found by 52 – 28
38
6.25, found by 50/8
The difference between 28 and 52 is 24. On the average the number of students enrolled
deviates 6.25 from the mean of 38.
37.
a.
b.
c.
d.
30, found by 54 – 24
38, found by 380/10
7.2, found by 72/10
The difference between 54 and 24 is 30. On the average the number of minutes required
to install a door deviates 7.2 minutes from the mean of 38 minutes.
38.
a.
b.
c.
d.
7.6%, found by 18.2 – 10.6
13.85% found by 110.8/8
2%, found by 16/8
The difference between 18.2 and 10.6 is 7.6%. On the average the return on investment
deviates two percent from the mean of 13.85%.
39.
a.
b.
c.
d.
The ranges are 32, found by 46 – 14, and 19, found by 35 – 16, respectively.
The means are 33.1, found by 331/10, and 24.5, found by 245/10, respectively.
The mean deviations are 7.88 and 4.20, respectively.
The California group is more diverse, but it does have a higher mean.
40.
a.
b.
c.
d.
The ranges are 10 and 5, respectively.
The means are 3.5 and 1.13, respectively.
The mean deviations are 2.38 and 1.19, respectively.
The Chickpee employees have both fewer lost days and less randomness.
41.
a.
5
b.
4.4 found by
a.
8
b.
9.67 found by
a.
$2.77
b.
2 
42.
43.
(8  5)2  (3  5) 2  (7  5) 2  (3  5) 2  (4  5) 2
5
(13  8)2  (3  8)2  (8  8)2  (10  8)2  (8  8)2  (6  8)2
6
(2.68  2.77) 2  ...  (4.30  2.77) 2  (3.58  2.77) 2
 1.26
5
29
Chapter 3
44.
a.
b.
11.76%, found by 58.8/5
16.89, found by 84.452/5
45.
a.
Range = 7.3, found by 11.6 – 4.3
Arithmetic mean = 6.94, found by 34.7/5
Variance = 6.5944, found by 32.972/5
Standard Deviation = 2.568
Dennis has a higher mean return (11.76 > 6.94). However, Dennis has greater spread in
their returns on equity (16.89 > 6.59).
b.
46.
a.
b.
c.
d.
$18,000, found by $140,000 – $122,000
$129,600, found by $648,000/5
Variance = 40,240,000, found by 201,200,000/5
Standard Deviation = $6343.50
Means about the same, but less dispersion in salary for TMV vice presidents.
47.
a.
X 4
b.
s = 2.3452
a.
X 8
b.
s = 2.3452
a.
X  38
b.
s = 9.0921
a.
X  13.85
b.
s = 2.4512
a.
X  95.1
b.
11.12
48.
49.
50.
51.
52.
a. X  104.1
b.
s2 
s2 
(7  4) 2  ...  (3  4) 2
 5.5
5 1
s2 
(11  8)2  ...  (7  8)2
 5.5
5 1
s2 
(28  38)2  ...  (42  38) 2
 82.6667
10  1
s2 
s2 
(10.6  13.85)2  ...  (15.6  13.85) 2
 6.0086
8 1
(101  95.1) 2  (97  95.1) 2 
10  1
(110  104.1) 2  (126  104.1) 2 
10  1
 (88  95.1) 2
 (100  104.1) 2
 123.66
 120.78
10.99
1
(1.8) 2
53.
About 69%, found by 1 
54.
About 84%, each income levels lie 2.5 standard deviations from the mean. Then
1
1
 0.84
(2.5)2
55.
a.
b.
Chapter 3
About 95%
47.5%, 2.5%
30
56.
87.23 and 96.57, found by 91.9  4.67 on 68 percent of the days; 82.56 and 101.24, found by
91.9  2(4.67) on 95 percent of the days.
57.
Because the exact values in a frequency distribution are not known, the midpoint of the class is
used for every member of that class.
58.
Class
f
M
fM
0 up to 5
2
2.5
5.00
5 up to 10
7
7.5
52.50
10 up to 15
12
12.5 150.00
15 up to 20
6
17.5 105.00
20 up to 25
3
22.5
67.50
Total
30
380.00
380
824
X
 12.67 s 
 5.33
29
30
59.
Class
f
20 up to 30
7
30 up to 40
12
40 up to 50
21
50 up to 60
18
60 up to 70
12
Total
70
3310
X
 47.2857
70
60.
Age
f
10 up to 20
3
20 up to 30
7
30 up to 40
18
40 up to 50
20
50 up to 60
12
Total
60
2410
X
 40.17
60
61.
Amount
f
20 up to 30
1
30 up to 40
15
40 up to 50
22
50 up to 60
8
60 up to 70
4
Total
50
2240
X
 44.8
50
f ( M  x) 2
207
187
0
140
290
824
f ( M  x) 2
M
175
3477
420
1811
945
110
990
1071
780
3766
3310
10,234
10,234
s
 12.179
69
X
25
35
45
55
65
M
15
25
35
45
55
f ( M  x) 2
X
25
35
45
55
65
f ( M  x) 2
fM
45
175
630
900
660
2410
7098
s
 10.97
60  1
M
25
525
990
440
260
2240
4298
s
 9.37
50  1
1901
1611
481
467
2639
7098
392
1441
1
832
1632
4298
31
Chapter 3
62.
63.
64.
65.
66.
Expenditure
25 up to 35
35 up to 45
45 up to 55
55 up to 65
65 up to 75
Total
3120
X
 52
60
a.
f
5
10
21
16
8
60
f ( M  x) 2
M
150
2420
400
1440
1050
84
960
1024
560
2592
3120
7560
7560
s
 11.32
60  1
X
30
40
50
60
70
b.
c.
Mean = 5, found by (6 + 4 + 3 + 7 + 5)/5
Median is 5, found by rearranging the values and selecting the middle value.
Population because all partners were included.
( X   )  (6  5)  (4  5)  (3  5)  (7  5)  (5  5)  0
a.
b.
Mean = 21.71, Median = 22.00
(23  21.7)  (19  21.7)  ...  (22  21.7)  0
545
 34.06
16
2116
X
 70.5333
30
X
Median = 37.50
67.
The Communications industry has older workers than the Retail Trade. Production
workers have the most age difference.
68.
a.
b.
c.
69.
Xw 
$5.00(270)  $6.50(300)  $8.00(100)
 $6.12
270  300  100
70.
Xw 
3(4)  3(4)  5(3)  2(3)  1(4)
 3.50
3  3  5  2 1
71.
Xw 
[15,300(4.5)  10, 400(3.0)  150,600(10.2)]
 9.28
176,300
72.
a.
b.
c.
73.
GM =
Chapter 3
4.84, found by 121/25
Median = 4.0
On half the days she made at least 4 appointments. The arithmetic mean number of
appointments per day is 4.84.
$3.438 found by 51.57/15
$3.44
$3.49
21
6, 286,800
 1  1.0094, so about 0.94%
5,164,900
32
74.
GM = 10
33,598
 1  1.0300  1  0.03 or 3.0 percent
25,000
GM = 10
44,771
 1  1.0599995  1  0.06 or 6.0 percent
25,000
75.
a.
b.
c.
55, found by 72 – 17
14.4, found by 144/10 where X = 43.2
17.6245
76.
a.
b.
c.
9, found by 12 – 3
2.72, found by 13.6/5 where mean = 7.6
3.5071
77.
a.
b.
population
183.47
78.
f×M M–
(M-)^2
f(M-)^2
42
-10.96
120.122
1681.70
378
-4.96
24.602
1033.27
870
1.04
1.082
62.73
588
7.04
49.562
1387.72
216
13.04
170.042
1360.33
2094
5525.8
The mean is 13.96, found by 2094/150. The standard deviation is 6.07, found by the square root
of 5525.8/150.
f
14
42
58
28
8
79.
a.
b.
c.
M
3
9
15
21
27
The times are a population because all flights are included.
The mean is 173.77, found by 2259/13. The median is 195.
The range is 294, found by 301 – 7.
X
9
195
241
301
216
260
7
244
192
147
10
295
142
X–
–164.769
21.231
67.231
127.231
42.231
86.231
–166.769
70.231
18.231
–26.769
–163.769
121.231
–31.769
(X – )^2
27148.9
450.7
4520.0
16187.7
1783.4
7435.7
27812.0
4932.4
332.4
716.6
26820.4
14696.9
1009.3
133846
The standard deviation is 101.47, found by the square root of 133846/13.
33
Chapter 3
80.
a.
b.
c.
The times are a population because all tables for that night are included.
The mean is 40.84, found by 1021/25. The median is 39.
The range is 44, found by 67 – 23. The standard deviation is 14.55, found by the square
root of 5291.4/25.
81.
a.
b.
The mean is 9.1, found by 273/30. The median is 9.
The range is 14, found by 18 – 4. The standard deviation is 3.566, found by the square
root of 368.7/29.
There are 30 customers. So 5 classes are appropriate. (18-4)/5 = 2.8 and an interval of 3
is used.
c.
Class
3.5 up to 6.5
6.5 up to 9.5
9.5up to 12.5
12.5 up to 15.5
15.5 up to 18.5
d.
Class
80 up to 100
100 up to 120
120 up to 140
140 up to 160
160 up to 180
180 up to 200
83.
f
10
6
9
4
1
f(M)
50
48
99
56
17
270
The mean is $141.20, found by 7060/50.
The standard deviation is $26.24, found by the square root of 33728/49.
The empirical rule says that 95 percent of the costs are between $88.72 and 193.68, found
by 141.20  2(26.24).
1807.5
 5.118
69
84.
Answers will vary.
85.
Answers will vary.
86.
Answers will vary.
Chapter 3
f(M – )^2
160
6
36
100
64
366
M
f f(M) M –  (M – )^2 f(M – )^2
90 3 270 –51.2 2621.44
7864.3
110 8 880 –31.2
973.44
7787.5
130 12 1560 –11.2
125.44
1505.3
150 16 2400
8.8
77.44
1239.0
170 7 1190 28.8
829.44
5806.1
190 4 760
48.8
2381.44
9525.8
7060
33728
Mean is 13, found by 910/70
s
M –  (M – )^2
-4
16
-1
1
2
4
5
25
8
64
Now the mean appears to be 9, found by 270/30. The standard deviation is 3.552, found
by the square root of 366/29. We are not surprised at the slight difference because the
midpoints may be different from the actual values in a class.
82.
a.
b.
c.
M
5
8
11
14
17
34
87.
a.
1. A software package gave the output:
Descriptive Statistics: Price
Variable
Price
Variable
Price
N
105
Mean
221.10
Minimum
125.00
Median
213.60
Maximum
345.30
TrMean
220.00
Q1
186.85
StDev
47.11
SE Mean
4.60
Q3
251.85
2. The distribution is symmetric about $220,000 with most of the prices less than
$30,000 away from the center.
b.
1. A software package gave the output:
Descriptive Statistics: Size
Variable
N
Mean
Median
TrMean
StDev
Size
2218.9
248.7
Variable
Size
105
2223.8
Minimum
1600.0
2200.0
Maximum
2900.0
Q1
2100.0
SE Mean
24.3
Q3
2400.0
2. The distribution is symmetric about 2200 square feet with most of the sizes less than
200 square feet away from the center.
88.
a.
b.
c.
89.
a.
b.
The mean is $73,063,563. The median is $66,191,417.
The standard deviation is $34,233,970.
In 2006, the mean age is 25.20. The median age is 14.50.
The standard deviation is 25.94.
The mean is 45,913. The median is 44,174.
The standard deviation is 5894.
1.
2.
The mean is 73.81; median = 76.10 and the standard deviation is 6.90
The distribution is quite negatively skewed with the coefficient of skewness
equal to 2.10027.
Select the variable GDP/cap
1.
The mean is 16.58; median = 17.45 and the standard deviation = 9.27.
2.
The distribution is fairly symmetric with the coefficient of skewness equal to
0.0567462.
35
Chapter 3