Download CHAPTER 3

CHAPTER 3
DESCRIBING DATA: NUMERICAL MEASURES
1.
  5.4 found by 27/5
2.
  55
. 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
(13
.  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.
$22.91, found by
12.
$1.50 found by ($40 + $35)/50
13.
$11.50, found by ($400 + $500 + $1400)/200
14.
$143.75, found by ($1000 +$750 + $4000)/40
15.
a.
b.
c.
16.
Median = 33,
Mode = 15
17.
Median = 5
Mode = 5
18.
Median = 10.5 Mode = 8
Chapter 3
300($20)  400($25)  400($23)
300  400  400
no mode
The given value would be the mode
3 and 4, bimodal
24

$25, 200
1100
19.
a.
b.
Median = 71.75%
Mode = 69.4%
20.
Median = 9.2 Modes are 8.2, 8.5, and 10.3
21.
12.8% increase found by
22.
6% increase found by
23.
12.28% increase found by
5
(1.094)(1.138)(1.117)(1.119)(1.147)  1
24.
54.48% increase found by
3
2, 400,000,000
1
651,000,000
25.
10.33% increase found by
26.
GM = 10
27.
10.76% found by
28.
5.49% for public colleges found by
29.
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 service reps on duty deviates by 2.4 from the mean of 6.
30.
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.
31.
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.
8
5
(1.08)(1.12)(1.14)(1.26)(1.05)  1
(1.02)(1.08)(1.06)(1.04)(1.10)(1.06)(1.08)(1.04)  1
13
14.0
1
3.9
54.87
 1  0.1956
9.19
5
70
1
42
8954
1
4975
22,608
1
5.70% for private colleges found by 11
12, 284
The rates of increase are about the same with private colleges slightly higher.
11
25
Chapter 3
32.
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%.
33.
a.
b.
c.
d.
15, found by 41 – 26
33.9, found by 339/10
4.12, found by 41.2/10
The ratings deviate 4.12 from the mean of 33.9 on the average.
34.
a.
b.
c.
d.
10 days, found by 10 – 0
3.5 days found by 28/8
2.375 days, found by 19/8
Days lost to illness deviates 2.375 days on average from the mean. The difference of 10
and 0 is 10.
35.
a.
5
b.
4.4 found by
a.
8
b.
9.67 found by
a.
$2.77
b.
2 
38.
a.
b.
11.76%, found by 58.8/5
16.89, found by 84.452/5
39.
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).
36.
37.
b.
(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  (358
.  2.77) 2
 126
.
5
40.
a.
b.
c.
d.
$18,000, found by $90,000 – 72,000
$79,600, found by $398,000/5
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.
41.
a.
X 4
s2 
(20) 2
5  5.50
5 1
102 
b.
s2 
c.
s = 2.3452
Chapter 3
(7  4)2 ...(3  4)2
 5.5
5 1
26
42.
43.
44.
45.
a.
X 8
s2 
(11  8)2 ...(7  8)2
 5.5
5 1
(40)2
5  5.50
5 1
342 
b.
s2 
c.
s = 2.3452
a.
X  38
s2 
(28  38)2 ...(42  38)2
 82.6667
10  1
(380) 2
10  82.6667
10  1
15,184 
b.
s2 
c.
s = 9.0921
a.
X  13.85
s2 
(10.6  13.85)2 ...(15.6  13.85)2
 6.0086
8 1
(110.8) 2
8
 6.0086
8 1
1576.64 
b.
s2 
c.
s = 2.4512
a.
X  124
s2 
(124  124)2 ...(121  124)2
 4.6667
10  1
(1240) 2
10  4.6667
10  1
153,802 
b.
s2 
c.
s = 2.1602
46.
AB4 yields a higher mean weight with less spread.
47.
a.
b.
About 95%
47.5%, 2.5%
48.
a.
b.
c.
d.
85, halfway between the endpoints of 140 and 30
About 18, found by (140 – 30)/6
103 and 67, found by 85  (1)18
121 and 49, found by 85  (2)18
49.
8.06%, found by (0.25/3.10)(100)
50.
Domestic 23.81%, found by (5/21)(100). Overseas 20%, found by (7/35)(100).
There is slightly more relative dispersion in the weights of luggage for domestic passengers.
51.
a.
b.
Because the two series are in different units of measurement.
P.E. ratio = 16.51%
ROI 20.8%, less spread in the P.E. ratios
52.
a.
b.
The data are in the same units but the means, relatively speaking, are far apart.
The relative dispersion in stocks under $10 is 28.95%. For stocks over $60, 5.71%. Less
relative dispersion in stocks over $60.
27
Chapter 3
53.
a.
The mean is 30.8, found by 154/5. The median is 31.0 and the standard deviation is 3.96,
4806 
found as
b.
54.
a.
154 2
5 .
4
3(30.8  31.0)
0.15, found by
.
3.96
The mean is 542, found by 8130/15. The median is 546 and the standard deviation is
4,415,268 
25.08, found as
b.
55.
a.
b.
56.
a.
14
3(542  546)
–0.478, found by
25.08
The mean is 21.93, found by 328.9 / 15. The median is 15.8 and the standard deviation is
328.9 2
13,494.676 
15 .
21.18, found as
14
3(21.93  15.8)
0.868, found by
.
21.18
The mean is 3.046, found by 88.34/29. The median is 2.00 and the standard deviation is
514.9526 
2.963, found as
b.
8130 2
15 .
28
3(3.046  2)
1.059, found by
2.963
88.34 2
29 .
57.
Median = 53 found by (11 + 1)(1/2) therefore 6th value in from lowest.
Q1 = 49 found by (11 + 1)(1/4) therefore 3rd value in from lowest
Q3 = 55 found by (11 + 1)(3/4) therefore 9th value in from lowest
58.
Median = 9.53, found by (9.45 + 9.61)/2
Q1 = 7.69 found by 7.59 + (7.99 – 7.59) ¼
Q3 = 12.59 found by 12.22 + (12.71 – 12.22)3/4
59.
a.
b.
c.
Q1 = 33.25
D2 = 27.8
P67 = 47
60.
a.
b.
c.
d.
Median = 58
Q1 = 51.25
D1 = 45.3
P33 = 53.53
Chapter 3
Q3 = 50.25
D8 = 52.6
Q3 = 66.0
D9 = 76.4
28
61.
a.
b.
c.
d.
e.
f.
350
Q1 = 175
Q3 = 930
755, found by 930 – 175
Less than zero, or more than about 2060
There are no outliers
The distribution is positively skewed
62.
a.
b.
c.
d.
e.
f.
450
Q1 = 300
Q3 = 700
400, found by 700 – 300
Less than zero or more than 1300
One outlier at about 1500
Distribution is positively skewed
63.
The distribution is somewhat positively skewed. Note that dashed line above 15.5 is longer than
below 7.8.
BoxPlot
0
5
10
15
20
25
64.
The median is $253, the smallest value is $116 and the largest is $353. About 25% of the semiprivate rooms are less than $214 and 25% above $304. The distribution is negatively skewed.
65.
Because the exact values in a frequency distribution are not known, the midpoint of the class is
used for every member of that class.
29
Chapter 3
66.
Class
0 up to 5
5 up to 10
10 up to 15
15 up to 20
20 up to 25
Total
X
67.
68.
70.
X
25
35
45
55
65
s
M
15
25
35
45
55
2410
 40.17
60
s
f
1
15
22
8
4
50
2240
 44.8
50
Expenditure
25 up to 35
35 up to 45
45 up to 55
55 up to 65
Chapter 3
f
7
12
21
18
12
70
f
3
7
18
20
12
60
Amount
20 up to 30
30 up to 40
40 up to 50
50 up to 60
60 up to 70
Total
X
M
2.5
7.5
12.5
17.5
22.5
s
3310
 47.2857
70
Age
10 up to 20
20 up to 30
30 up to 40
40 up to 50
50 up to 60
Total
X
69.
380
 12.67
30
Class
20 up to 30
30 up to 40
40 up to 50
50 up to 60
60 up to 70
Total
X
f
2
7
12
6
3
30
X
25
35
45
55
65
s
f
5
10
21
16
X
30
40
50
60
fM
5.00
52.50
150.00
105.00
67.50
380.00
fM2
12.50
393.75
1875.00
1837.50
1518.75
5637.50
5637.5  (380)2 / 30
 5.33
29
M
175
420
945
990
780
3310
fM2
4375
14,700
42,525
54,450
50,700
166,750
166,750  (3310)2 / 70
 12.179
69
fM
45
175
630
900
660
2410
fM2
675
4375
22,050
40,500
36,300
103,900
103,900  (2410)2 / 60
 10.97
60  1
M
25
525
990
440
260
2240
fM2
625
18,375
44,550
24,200
16,900
104,650
104,650  (2240) 2 / 50
 9.37
50  1
M
fM2
150
4500
400 16,000
1050 52,500
960 57,600
30
65 up to 75
Total
X
71.
8
60
3120
 52
60
s
560 39,200
3120 169,800
169,800  (3120)2 / 60
 11.32
60  1
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
72.
a.
b.
Mean = 21.71, Median = 22.00
(23  21.7)  (19  21.7) ... (22  21.7)  0
73.
X
74.
a.
70
545
 34.06
16
2116
X
 70.5333
30
Median = 37.50
75.
370.08, found by 18,504/50
76.
a.
b.
c.
77.
Xw 
$5.00(270)  $6.50(300)  $8.00(100)
 $6.12
270  300  100
78.
Xw 
3(4)  3(4)  5(3)  2(3)  1(4)
 3.50
3  3  5  2 1
79.
Xw 
[15,300( 4.5)  10,400(3.0)  150,600(10.2)]
 9.28
176,300
80.
a.
b.
c.
arithmetic mean = 6.49 km
median = 6.3 km
mode = 5.3 and 4.6 km (bimodal)
81.
GM =
21
82.
GM =
10
GM =
10
a.
b.
55, found by 72 – 17
14.4, found by 144/10 where X = 43.2
83.
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.
6,286,800
 1  1.0094, so about 0.94%
5,164,900
33,598
 1  1.0300  1  0.03 or 3.0 percent
25,000
44,771
 1  1.0599995  1  0.06 or 6.0 percent
25,000
31
Chapter 3
c.
17.6245
84.
a.
b.
c.
9, found by 12 – 3
2.72, found by 13.6/5 where mean = 7.6
3.5071
85.
a.
b.
c.
population
183.47
94.92%
86.
a.
30 found by 30 – 0
(2094) 2
150
149
34,758 
b.
6.09 found by
87.
The following results were found using MINITAB
Q1 = 44.25
Q3 = 68.50
Median = 55.50
The distribution is approximately symmetric. The box plot is as follows.
-------------------------------------------|
+
|-----------------------------------|----------|----------|----------|----------|----------|--24
36
48
60
72
84
88.
a.
b.
c.
---------------------+
------------------------------------|-----|-----|-----|-----|-----|-----|-----|
10
15
20
25
30
35
40
45
No outliers
The distribution is positively skewed. The median time to change a muffler is 23.50
minutes. The first quartile is 15.75 minutes and the third quartile is 29.25 minutes. The
range of time is 10 minutes to 44 minutes.
89.
The distribution is positively skewed. The first quartile is approximately $20 and the third
quartile is approximately $90. There is one outlier located at $255. The median is about 50.
90.
The distribution is positively skewed. The first quartile is equal to 10 and the third quartile is
equal to 40. There are four outliers located at 85, 86, 95 and 99. The median is about 25.
91.
a.
b.
c.
d.
Chapter 3
857.90
Median = 16.35
 17158
.
50
(857.90)2
20,206.73 
50
s
 10.58
50  1
17158
.
 (2)(10.58)  4.002 and 38.318
10.58
CV 
(100)  61.66 %
17158
.
X
32
e.
f.
92.
3(17158
.
 16.35)
 0.23
10.58
25
75
Lp  (50  1)
 12.75
Lp  (50  1)
 38.25
100
100
Q1 = 7.825
Q3 = 27.400
---------------------------|
+
|----------------------------------+----------+---------+----------+----------+----0
10
20
30
40
sk 
g.
The distribution is nearly symmetrical. The mean is 17.158, the median is 16.35 and the
standard deviation is 10.58. About 75 percent of the companies have a value less than
27.4 and 25 percent have a value less than 7.825.
a.
Computer Output:
Variable
Assets
5yr
1yr
Variable
Assets
5yr
1yr
N
20
20
20
Mean
42220
145.8
-0.27
Minimum
22742
41.4
12.90
Median
36018
132.7
-1.60
Maximum
104357
264.3
13.20
TrMean
39850
145.0
-0.32
Q1
29097
119.8
-4.20
StDev
22495
48.8
6.10
SE Mean
5030
10.9
1.36
Q3
45484
184.9
2.88
Five-year returns are more variable.
b.
(22,495/42,220)(100) = 53.3% for assets;
(48.8/145.8)(100) = 33.5% for five-year returns;
(6.10/0.27)(100) = 2259.3% for one-year returns.
The relative variation of one-year returns is quite high.
c.
Software gives the following coefficients of skewness:
Assets 2.170968 Five-year
0.45416 One-year
0.484715
Assets are definitely positively skewed, but the other two variables are symmetric.
d.
See printout above.
e.
The box plot for five-year rates of return is:
-------------------|
+
|---------------------------+----------+---------+-------50
150
250
The box plot for one-year rates of return is:
33
Chapter 3
----------------|
+
|-------------------+----------+---------+-----10
0
10
93.
a.
The mean is 173.77 hours, found by 2259/13. The median is 195 hours.
s = 105.61 hours, found by
b.
c.
d.
2259 2
526,391 
13 .
12
 105.61 
CV = 60.78%, found by 
 100
 173.77 
Coefficient of skewness is –0.697
L45 = (14)(.45) = 6.3. So the 45th percentile is 192 + 0.3(195192) = 192.9.
L82 = (14)(.82) = 11.48. So the 82nd percentile is 260 + 0.48(295260) = 276.8.
----------------------|
+
|------------------------------+------+------+------+------+------+---0
75
150
225
300
375
There is a slight negative skewness visible, but no outliers.
94.
a.
b.
95.
X  96.55 found by 10,620/110
s
1,029,937.5 
109
(10,620) 2
110
 6.514
Mean is 13, found by 910/70
s
96.
a.
b.
13,637.50 
69
(910) 2
70  5118
.
mean = $2706.67; median = $2235
BoxPlot
0
1000
2000
3000
4000
5000
6000
#1
There is one extreme outlier. This observation will distort the value of the mean.
Chapter 3
34
7000
c.
97.
The median will be more representative of the data than the mean due to the high
extreme outlier.
The following output is from MegaStat.
a.
Mean
sample variance
sample standard deviation
Minimum
Maximum
Range
437,595.96
69,666,712,637
263,944.53
137000
1600000
1463000
Skewness
Kurtosis
coefficient of variation (CV)
1.52
4.14
60.32%
1st quartile
Median
3rd quartile
interquartile range
Mode
200,000
435,000
574,000
374,000
200,000
1. see the above output.
2. see the above output. The distribution is positively skewed.
3.
BoxPlot
0
200000
400000
600000
800000
1000000 1200000
1400000 1600000 1800000
List price
There are two mild outliers. From the boxplot, the 1st and 3rd quartiles are about 200 000 and 575 000.
4. The outliers will distort the value of the mean, and so the median is more
representative of the data.
b.
Count
Mean
sample variance
sample standard deviation
Minimum
Maximum
99
2,096.91
1,447,318.23
1,203.05
196
10600
35
Chapter 3
Range
10404
Skewness
Kurtosis
coefficient of variation (CV)
1st quartile
Median
3rd quartile
interquartile range
Mode
3.89
24.90
57.37%
1,356.00
1,946.00
2,479.50
1,123.50
1,400.00
1. see the above output
2. see the above output. The distribution is positively skewed.
3.
BoxPlot
0
2000
4000
6000
8000
10000
12000
Size (in sq ft)
There are two mild and one extreme outlier. From the boxplot, the 1st and 3rd quartiles are about 1400 and
2500.
4. The outliers will distort the value of the mean, and so the median is more
representative of the data.
98.
a.
Chapter 3
1.
2.
The mean is 65.49; median is 64.0 and the standard deviation is 24.87.
The coefficient of skewness is positive, namely 0.289022.
36
3.
There are no outliers. The first quartile is near 43 and the third quartile is about 86.
20
30
40
50
60
70
80
90
100
110
120
Salary
b.
4.
The distribution is fairly symmetric around 65. It seems to have large
“shoulders” at the quartiles and then fall off very quickly.
1.
2.
3.
The mean is 24.13; median = 14.0 and the standard deviation is 24.72.
The coefficient of skewness is quite positive, namely 1.52649.
There are two outliers at 88 and 90. The first quartile is near 6 and the third is 36.
0
10
20
30
40
50
60
70
80
90
Age
c.
4.
The age distribution is quite skewed with a typical value of 14 years and most of
the stadiums are between 5 and 40 years of age.
1.
2.
3.
The mean is 46,947; median = 46,056 and the standard deviation is 6475.
The coefficient of skewness is positive, namely 0.330510.
There are no outliers. The first quartile is near 42,000 and the third quartile is
near 51,000.
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Chapter 3
30000
40000
50000
60000
Size
4.
99.
a.
1.
2.
3.
The distribution is quite symmetric around 46,500 with the most values within
5000 of the center.
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.
There are two outliers at 48 and 51. The first quartile is near 72 and the third
quartile is near 78.
50
60
70
80
Life Expectancy
4.
b.
Chapter 3
The life expectancy distribution is heavily concentrated near 75 with a slight
negative tail into the 60’s and , of course, the two outliers near 50.
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.
3.
There are no outliers. The first quartile is about 8 and the third quartile is near 24.
38
0
10
20
30
40
GDP/cap
4.
The distribution of GDP/capita is symmetric about 16 and most of the values are
within 8 of that center.
39
Chapter 3