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Chapter 03 - Descriptive Statistics: Numerical Methods
CHAPTER 3—Descriptive Statistics: Numerical Methods
3.1
3.2
a.
Population parameter: number calculated using the population measurements that
describe some aspect of the population.
Point estimate: a one-number estimate of the value of a population parameter.
b.
The sample mean is the point estimate of the population mean.
a.
Equal or roughly equal.
b.
Mean is less than the median.
c.
Mean is greater than the median.
3.3
a
b
N
10
9
MEAN
9.600
103.33
MEDIAN
10.000
100.00
MODE
10.00
90.00
a
b
N
10
7
MEAN
20
503
MEDIAN
20
501
MODE
20
501
3.4
3.5
3.6
3.7
3.8
a.
Sample mean = 42.954. Yes, because x  42 .
b.
The sample median is 43.0
The mean of 42.954 is slightly less than the median of 43.
The histogram is slightly skewed to the left so the mean should be less than the median.
a.
Sample mean = 5.46. Yes, because x  6 .
b.
Sample median = 5.25
The mean is slightly larger than the median. The histogram is somewhat skewed right
which is consistent with the mean being larger than the median.
a.
Sample mean = 50.575. Yes, because x  50.
b.
Sample median = 50.65. The mean and median are very similar which is consistent with
the histogram being close to symmetric.
Mean = 99 and the median = 40. Median is better measure of typical because of the high
outlier (350).
3-1
Chapter 03 - Descriptive Statistics: Numerical Methods
3.9 – 3.13 MINITAB output for all variables on data comparing lifestyles.
a.
Variable
voters
income tax
video reentals
PCs
Religion
b.
3.9
3.10
3.11
3.12
3.13
c.
N
9
9
8
8
6
Mean
70.70
49.44
4.03
18.94
31.65
Median
71.50
50.00
2.10
17.50
26.25
StDev
11.81
7.30
4.49
7.10
18.21
Min
49.1
40.0
0.7
11.5
10.0
Max
85.0
60.0
13.8
35.0
55.8
Q1
61.35
42.00
1.13
14.50
17.50
Somewhat skewed left; the U.S. has the lowest percentage.
Slightly skewed left, the U.S. has the lowest percentage, tied with Britain.
Skewed right; U.S. is the highest.
Skewed right; U.S. is the highest.
Skewed right; U.S. is above the mean and median.
All dot plots below show the same skewness as noted above.
DotPlot
40
50
60
70
80
90
Voters
DotPlot
30
40
50
60
70
Income Tax
DotPlot
0
5
10
Video Rentals
3-2
15
Q3
80.40
55.50
6.45
20.00
52.65
Chapter 03 - Descriptive Statistics: Numerical Methods
DotPlot
10
15
20
25
30
35
40
40
50
60
PCs
DotPlot
0
10
20
30
Religion
d.
3.14
U.S. has the lowest percentage.
U.S. has the lowest percentage, tied with Britain.
U.S. is the highest.
U.S. is the highest.
U.S. is above the mean and median.
Sample mean = 83.6 and sample median = 78.5 indicating data is skewed right.
3.15 a.
3.16
3.9
3.10
3.11
3.12
3.13
Skewed to the right. Most players earned the league minimum.
b.
Earning more than the mean: about 33%
Earning more than the median: 50%
c.
Explanations will vary. All quotes are a distortion/misrepresentation of the true earnings by
the majority of players.
The range is the difference between the largest and smallest measurements in a population or
sample. It is a simple measure of variation.
The variance is the average of the squared deviations from the average. It measures variability
in square units.
The standard deviation is the square root of the variance.
3.17
Both the variance and standard deviation measure the spread of the individual values about the
mean. The larger the spread, the more variation in the data.
3-3
Chapter 03 - Descriptive Statistics: Numerical Methods
3.18
The variance and standard deviation are calculated by using all of the data, while the range is
calculated using only the largest and smallest values.
3.19
range = 15 – 5 = 10
5
(x
i
–  )2
(5 – 10) 2  (8 – 10) 2  (10 – 10) 2  (12 – 10) 2  (15 – 10) 2
5
5
25  4  0  4  25

 11.6
5
  11.6  3.4059
2 
i 1

3.20
Housing Affordability in Texas
x
57.5
61.7
32.5
67.4
55.7
49.2
(x – 54)
3.5
7.7
–21.5
13.4
1.7
-4.8
(x – 54)**2
12.25
59.29
462.25
179.56
2.89
23.04
sum(x – 54)**2 PopVariance PopStDev
739.28
123.21
11.10
Mean = 54
Range = 67.4 – 32.5 = 34.9
Variance = 123.21
Std Deviation = 11.10
3-4
Chapter 03 - Descriptive Statistics: Numerical Methods
3.21
a
Airline Revenues
Revenue
$ Billions
(Rev - 11.43)^2
22.6
19.3
17.2
13.1
12.6
11.6
9.1
3.3
3.1
2.4
124.7689
61.9369
33.2929
2.7889
1.3689
0.0289
5.4289
66.0969
69.3889
81.5409
114.3
446.641
Mean =
11.43
Variance = 446.641 / 10 =
44.664
Sum
Std. Dev. = Sqrt (Variance)=
6.683
Range = 22.6 – 2.4 =
20.200
Airline Profits
Profits
$ millions
(Rev - 1530.7)^2
231
22,876
-6,203
343
-2,835
304
499
-53
146
-1
1689220.09
455621832.1
59810115.69
1410631.29
19059336.49
1504792.89
1064404.89
2508105.69
1917394.09
2346104.89
Sum
15307
546931938.1
Mean =
Variance =
1530.7
Std. Dev. =
Range =
54693193.810
7395.485
29079
3-5
Chapter 03 - Descriptive Statistics: Numerical Methods
b.
United Airlines is a high outlier and the only observation above the average. All other
airlines are within two standard deviations of the average.
Airline
American
Airlines
United Airlines
Delta Air Lines
Continental
Airlines
Northwest
Airlines
US Airways
Group
Southwest
Airlines
Alaska Air
Group
SkyWest
Jetblue Airways
3.22 a.
x
Profits
($
millions)
z-score
231
22,876
-6,203
-0.18
2.89
-1.05
343
-0.16
-2,835
-0.59
304
-0.17
499
-0.14
-53
146
-1
-0.21
-0.19
-0.21
157  132  109  145  125  139
 134.5
6
(157 – 134 .5) 2  (132 – 134 .5) 2  (139 – 134 .5) 2
s2 
= 276.7
6 –1
s  276.7  16.63
 x  157  132  109  145  125  139  109,925
 x   (157  132  109  145  125  139)  (807 )  651,249
2
i
Also
2
2
2
2
2
2
2
2
2
i
s2 
1 
n – 1 
x
2
i
–
1
n
 x    6 1– 1 109,925 – 16 (651,249)  276.7

2
i
b.
[ x  s]  [134.5  16.63]  [117.87,151.13]
[ x  2s]  [134.5  2(16.63)]  [101.24,167.76]
[ x  3s]  [134.5  3(16.63)]  [84.61,184.39]
c.
Yes, because $190 is not within the 99.73% interval.
d.
z157=
157  134.5
= 1.353
16.63
z132 = –0.150
z109 = –1.533
z145 = 0.631
3-6
Chapter 03 - Descriptive Statistics: Numerical Methods
z125 = –0.571
z139 = 0.271
Each expense is within 2 standard deviations of the mean.
3.23 a.
The histogram indicates that the breaking strengths are approximately normally distributed.
Therefore, the empirical rule is appropriate.
b.
68.26%: [50.575 ± 1(1.6438)] = [48.9312, 52.2188]
95.44%: [50.575 ± 2(1.6438)] = [47.2875, 53.8626]
99.73%: [50.575 ± 3(1.6438)] = [45.6436, 55.5064]
c.
We estimate that the breaking strengths of 99.73% of the trashbags are between 45.6436 lbs
and 55.5064 lbs. Thus, almost any trashbag will have a breaking strength that exceeds 45 lbs.
d.
27 out of 40 (or 67.5%) actually fall within the interval [ x  s].
38 out of 40 (or 95%) actually fall within the interval [ x  2s].
40 out of 40 (or 100%) actually fall within the interval [ x  3s].
These percentages compare favorably with those given by the Empirical Rule
(68.26%, 95.44%, and 99.73%). This suggests that our inferences are valid.
3.24 a.
It is somewhat reasonable.
b.
[ x  s]  [5.46  2.475]  [2.985,7.935]
[ x  2s]  [5.46  2(2.475)]  [.51,10.41]
[ x  3s]  [5.46  3(2.475)]  [–1.965,12.885]
c.
Yes, because the upper limit of the 68.26% interval is less than 8 minutes.
d.
66% fall into [ x  s], 96% fall into [ x  2s] , 100% fall into [ x  3s] .
Yes, they are reasonably valid.
3.25 a.
It is somewhat reasonable.
b.
[ x  s]  [42.95  2.6424 ]  [40.3076,45.5924 ]
[ x  2s]  [42.95  2(2.6424 )]  [37.6652,48.2348]
[ x  3s]  [42.95  3(2.6424 )]  [35.0228,50.8772 ]
c.
Yes, because the lower limit of the interval is greater than 35.
d.
63% fall into [ x  s], 98.46% fall into [ x  2s], 100% fall into [ x  3s] .
Yes, they are reasonably valid.
3-7
Chapter 03 - Descriptive Statistics: Numerical Methods
3.26 a.
Stem and Leaf plot for
stem unit =
leaf unit =
Time
10
1
Frequency
27
Stem
3
22
11
1
3
4
4
2
63
5
Leaf
222222222333333333334444444
555555566677777889999
9
00011122234
6
12
Histogram
35
30
20
15
10
5
Time
Distribution is skewed to the right.
3-8
54
52
50
48
46
44
42
40
38
36
34
0
32
Percent
25
Chapter 03 - Descriptive Statistics: Numerical Methods
b.
[ x  s]  [36.56  4.475]  [32.085,41.035] :73%
[ x  2s]  [36.56  2(4.475)]  [27.61,45.51] :95.23%
[ x  3s]  [36.56  3(4.475)]  [23.135,49.985] :96.83%
c.
They are inconsistent with the empirical rule, but consistent with Chebyshev’s Theorem.
d.
The transaction times are skewed to the right.
3.27 a.
RS Internet Age Fund:
Franklin Income A Fund:
Jacob Internet Fund:
[10.93 – 2*41.96, 10.93 + 2*41.96] = [–72.99, 94.85]
[13 – 2*9.36, 13 + 2*9.36] = [–5.72, 31.72]
[34.45 – 2*41.16, 34.5 + 2*41.16] = [–47.87, 116.77]
b.
RS has the lowest average and highest variability.
Franklin has the second lowest average and the smallest variability.
Jacob has the highest average return and the second highest variability.
c.
RS Internet Age Fund:
Franklin Income A Fund:
Jacob Internet Fund:
Coefficient of Variation = 41.96 / 10.93 * 100 = 383.9%
Coefficient of Variation = 9.36 / 13 * 100 = 72%
Coefficient of Variation = 41.14 / 34.45 * 100 = 119.4%
RS Internet is riskiest, Jacob is second and Franklin is least risky.
3.28 a.
The value such that a specified percentage of the measurements in a population or sample fall
at or below it.
b.
A value below which approximately 25% of the measurements lie
c.
The 75th percentile
d.
IQR  Q3  Q1 contains the middle 50% of the data.
3.29
In a box and whisker plot a point that is more than 1.5 times the IQR above Q3 or below Q1 is
considered to be an outlier.
3.30
Median = 8. Q3 = 9. Q1 = 7.5.
3-9
Chapter 03 - Descriptive Statistics: Numerical Methods
3.31 a.
192 b.
152
c.
141
d.
171
e.
132
f.
30
g.
Income
(1000's)
127
241
141.00
152.00
171.00
minimum
maximum
1st quartile
median
3rd quartile
BoxPlot
100
120
140
160
180
200
220
240
260
280
Income (1000's)
3.32 a.
Explanations will vary.
b.
High-tech sector is most variable. Retailing sector is least variable.
c.
High-tech and Insurance sectors
3.33
Overall, the thirty-year rates are higher than the 15-year rates. The variability is similar.
Average of the differences = .4444
3.34 a.
Q1 = 7.5, Q3 = 9
lower hinge = 7.5
upper hinge = 9
(7 + 8) / 2 = 7.5
(9 + 9) / 2 = 9
b.
Q1 = 138, Q3 = 177
lower hinge = 139.5 (138 + 141) / 2 = 139.5
upper hinge = 174 (171 + 177) / 2 = 174
c.
explanations will vary
3.35 a.
b.
All categories showed improvement.
Most: strategic quality planning, quality and operational results
Least: information and analysis, human resource development and management
3-10
Chapter 03 - Descriptive Statistics: Numerical Methods
c.
Increase: leadership, human resource development, customer focus and satisfaction
Decrease: strategic quality planning, management of process quality
Others stayed about the same.
d.
The skewness changed in leadership, strategic quality planning, human resource development,
and quality and operational results.
3.36
Both are measurements of linear association between two quantitative variables. If they are
positive, then the relationship is positive and if they are negative, then the relationship is
negative.
3.37
For a specific value of x you predict a value of y by plugging the value of x into the equation
of the least squares line.
3.38
sxy = -179.6475 / 7 = -25.66, sx = sqrt(1404.355 / 7) = 14.16, sy = sqrt(25.548 / 7) = 1.91,
r = -25.66 / (14.16 * 1.91) = -0.949, b1 = -25.66 / 14.162 = 0.1279, b0 = 10.2125 – (-0.1279) *
43.98 = 15.84
Predicted value at x = 40 degrees is 15.84 – 0.1279 * 40 = 10.724
3.39 a.
b.
There is a strong positive linear relationship between these two variables
Predicted service time is 11.4641 + 24.6022 * 5 = 134.4751 minutes.
3.40
Classess are weighted by credit hours. The xi values are the class grades and the weights are
the credit hours per class.
3.41
The midpoint equals the average of the measurements in the class.
3.42
The actual measurements are unknown.
3.43

100,000 (10.7)  500,000 (21.7)  500,000 (9.9)  200,000 (5.8)  50,000 (5.5)
100,000  500,000  500,000  200,000  50,000

18,305,000
 13.56%
1,350,000
a.
b.
unweighted mean = 10.72%
More money was invested in funds with larger gains.
3.44
a.

1.62(5.1)  5.09(7.1)  6.45(5.6)  3.18(5.4)  3.08(4.8)
1.62  5.09  6.45  3.18  3.08

112 .477
19.42
 5.79%
b.
unweighted mean = 5.6%
3-11
Chapter 03 - Descriptive Statistics: Numerical Methods
The weighted mean accounts for the different sizes of labor force.
3.45
3.46
2 

53(2)  118(5)  21(8)  3(11)
 4.6lbs
195
a.
x=
b.
53(2  4.6) 2  118(5  4.6) 2  21(8  4.6) 2  3(11  4.6) 2 742.8
s=

 3.829
195  1
194

2
1(24.5)  17(74.5)  5(124.5)  4(174.5)  1(224.5)  2(274.5) 3385

 112.83%
30
30
1(24 .5  112 .83) 2  17 (74 .5  112 .83) 2  5(124 .5  112 .83) 2  4(174 .5  112 .83) 2  1(224 .5  112 .83) 2  2(274 .5  112 .83) 2
20
113416 .667
 3780 .56
30
  3780 .56  61.49%
3.47
Midpoint
30
35
40
45
50
55
60
65
70
75
Freq
1
3
3
13
14
12
9
1
3
1
sample mean = 51.5
(M – mean)**2
462.25
272.25
132.25
42.25
2.25
12.25
72.25
182.25
342.25
552.25
M*f
30
105
120
585
700
660
540
65
210
75
variance = 81.61017
s = 9.033835
3-12
F*diff^2
462.25
816.75
396.75
549.25
31.5
147
650.25
182.25
1026.75
552.25
Chapter 03 - Descriptive Statistics: Numerical Methods
3.48
The geometric mean return for an investment is the constant return on the investment for the
investment period.
3.49
Yes, the ending value is I(1+Rg)n , where I is the initial investment.
3.50
a.
Rg  3 (1  .1)(1  .1)(1  .25)  1  1.0736  1  .0736  7.36%
b.
$5,000,000(1+.0736)3=$6,187,500
1,000,000(1+Rg)4=4,000,000
3.51
(1+Rg)4 = 4
(1+Rg) =
4
4
Rg = 1.4142 – 1
Rg = .4142
3.52
a.
b.
3.53
2000–01:
1283 .27  1455 .22
 11.8%
1455 .22
2001–02:
1154 .67  1283 .27
 10.0%
1283 .27
2002–03:
909 .03  1154 .67
 21.3%
1154 .67
2003–04:
1108 .48  909 .03
 21.9%
909 .03
2004–05:
1211 .92  1108 .48
 9.3%
1108 .48
c.
Rg= 5 (1  .118)(1  .1)(1  .213)(1  .219 )(1  .093)  1  .9640  1  .036
d.
$1,000,000(1 – .036)5 = $832,501
a.
1990–95:
1096  789
 .3891
7891
1995–2000:
Rg =
1534  1096
 .3996
1096
(1.3891)(1.3996)  1  .39434
3-13
Chapter 03 - Descriptive Statistics: Numerical Methods
x  1534
 .39434
1534
b.
x  $2,139
3.54
Times greater than 17 are high outliers and they should be investigated.
BoxPlot
0
5
10
15
20
25
30
Time
3.55
Explanations will vary.
3.56 a.
Fixed Annuities:
68.26%
99.73%
[ ± ] = [8.31 ± .54] = [7.77, 8.85]
[ ± 3] = [8.31 ± 3(.54)] = [6.69, 9.93]
Cash Equivalents:
68.26%
99.73%
[ ± ] = [7.73 ± .81] = [6.92, 8.54]
[ ± 3] = [7.73 ± 3(.81)] = [5.30, 10.16]
U.S. Treasury Bonds:
68.26%
99.73%
[ ± ] = [8.80 ± 5.98] = [2.82, 14.78]
[ ± 3] = [8.80 ± 3(5.98)] = [–9.14, 26.74]
U.S. Corporate Bonds:
68.26%
99.73%
[[ ± ] = [9.33 ± 7.92] = [1.41, 17.25]
[ ± 3] = [9.33 ± 3(7.92)] = [–14.43, 33.09]
Non U.S. Gov’t Bonds:
68.26%
99.73%
[ ± ] = [10.95 ± 10.47] = [.48, 21.42]
[ ± 3] = [10.95 ± 3(10.47)] = [–20.46, 42.36]
Domestic Large Cap Stocks:
68.26%
99.73%
[ ± ] = [11.71 ± 15.30] = [–3.59, 27.01]
[ ± 3] = [11.71 ± 3(15.30)] = [–34.19, 57.61]
International Equities:
68.26%
99.73%
[ ± ] = [14.02 ± 17.16] = [–3.14, 31.18]
[ ± 3] = [14.02 ± 3(17.16)] = [–37.46, 65.5]
3-14
Chapter 03 - Descriptive Statistics: Numerical Methods
Domestic MidCap Stocks
68.26%
99.73%
[ ± ] = [13.64 ± 18.19] = [–4.55, 31.83]
[ ± 3] = [13.64 ± 3(18.19)] = [–40.93, 68.21]
Domestic Small Cap Stocks
68.26%
99.73%
[ ± ] = [14.93 ± 21.82] = [–6.89, 36.75]
[ ± 3] = [14.93 ± 3(21.82)] = [–50.53, 80.39]
b. and c. are for fixed annuities; others follow similar methods. The results for b are obtained
by using Chebyshev’s Theorem for k=2 and k=3. The results for c are 3 standard deviations
above the mean and 3 standard deviations below the mean.
b.
At least 75%: [7.23, 9.39]; at least 88.89%: [6.69, 9.93]
c.
9.93, 6.69
d.
Domestic small cap stocks: 80.39, domestic midcap stocks: 68.21
e.
Fixed annuities: 6.69; cash equivalents: 5.30
f.
Least risky: fixed annuities
coefficient of variation = (.54/8.31)100 = 6.498
Riskiest: domestic small cap stocks
coefficient of variation = (21.82/14.93)100 = 146.149
3.57 a.
b.
3.58 a.
b.
3.59 a.
About 70%
410
538  3(41) = 538  123 = [415, 661]
Yes, it is beyond the interval.
Average Price with Pools = $276,056
Average Price without Pools = $226,900
Difference = $49,156
% Recouped = $49,156/32,500 = 151%
b.
The assumption that the only variable causing the difference in average price is whether
or not there is a pool is really suspect.
Internet Exercise
3.60
Analyses will vary
3-15