Devore J.L., Berk K.N. Modern Mathematical Statistics with Applications
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Table A.11 Critical Values for the Ryan–Joiner Test of Normal ity
808 Appendix Tables
Table A.12 Critical Values for the Wilcoxon Signed-Rank Test
Appendix Tables 809
Table A.13 Critical Values for the Wilcoxon Rank-Sum Test
810 Appendix Tables
Table A.14 Critical Values for the Wilcoxon Signed-Rank Interval
Appendix Tables 811
Table A.15 Critical Values for the Wilcoxon Rank-Sum Interval
812 Appendix Tables
Table A.16 b Curves for t Tests
Appendix Tables 813
Answers to
Odd-Numbered Exercises
Chapter 1
1. a. Houston Chronicle, Des Moines Register, Chicago
Tribune, Washington Post
b. Capital One, Campbell Soup, Merrill Lynch,
Prudential
c. Bill Jasper, Kay Reinke, Helen Ford, David Menendez
d. 1.78, 2.44, 3.50, 3.04
3. a. In a sample of 100 DVD players, what are the chances
that more than 20 need service while under warranty?
What are the chances that none need service while still
under warranty?
b. What proportion of all DVD players of this brand and
model will need service within the warranty period?
5. a. No, the relevant conceptual population is all scores of
all students who participate in the SI in conjunction
with this particular statistics course.
b. The advantage of randomly allocating students to the
two groups is that the two groups should then be fairly
comparable before the study. If the two groups perform
differently in the class, we might attribute this to the
treatments (SI and control). If it were left to students to
choose, stronger or more dedicated students might
gravitate toward SI, confounding the results.
c. If all students were put in the treatment group there
would be no results with which to compare the
treatments.
7. One could generate a simple random sample of all single
family homes in the city, or a stratified random sample
by taking a simple random sample from each of the ten
district neighborhoods. From each of the homes in the
sample the necessary data would be collected. This
would be an enumerative study because there exists a
finite, identifiable population of objects from which to
sample.
9. a. There could be several explanations for the variability
of the measurements. Among them could be measuring
error, (due to mechanical or technical changes across
measurements), recording error, differences in
weather conditions at time of measurements, etc.
b. This study involves a conceptual population. There is
no sampling frame.
11. 6 l 034
6 h 667899
7 l 00122244
7 h Stem ¼ tens
8 l 001111122344 Leaf ¼ ones
8 h 5557899
9l 03
9h 58
This display brings out the gap in the data:
There are no scores in the high 70’s.
13. a. 2 23 Stem units: 1.0
3 2344567789 Leaf units: .10
4 01356889
5 00001114455666789
6 0000122223344456667789999
7 00012233455555668
8 02233448
9 012233335666788
10 2344455688
11 2335999
12 37
13 8
14 36
15 0035
16
17
18 9
b. A representative value could be the median, 7.0.
c. The data appear to be highly concentrated, except for
a few values on the positive side.
d. No, there is skewness to the right, or positive
skewness.
e. The value 18.9 appears to be an outlier, being more
than two stem units from the previous value.
15. a.
Number
nonconforming Frequency
Relative
frequency
(Freq/60)
0 7 0.117
1 12 0.200
2 13 0.217
3 14 0.233
4 6 0.100
5 3 0.050
6 3 0.050
7 1 0.017
81
0.017
1.001
Doesn’t add exactly to 1 because relative
frequencies have been rounded
814
b. .917, .867, 1 .867 ¼ .133
c. The center of the histogram is somewhere around
2 or 3 and it shows that there is some positive
skewness in the data.
17. a. .375
b. .218
c. .242
d. The histogram is very positively skewed.
19. a. The number of subdivisions having no cul-de-sacs is
17/47 ¼ .362, or 36.2%. The proportion having at
least one cul-de-sac is 30/47 ¼ .638, or 63.8%.
y: Count Percent
0 17 36.17
1 22 46.81
2 6 12.77
3 1 2.13
5 1 2.13
N ¼ 47
.362, .638
b.
21. a.
The histogram is skewed right, with a majority of
observations between 0 and 300 cycles. The class
holding the most observations is between 100 and 200
cycles.
b.
c. .79
23.
The original distribution is positively skewed.
The transformation creates a much more symmetric,
mound-shaped histogram.
25. a.
The distribution is skewed to the right, or positively
skewed. There is a gap in the histogram, and what
appears to be an outlier in the ‘500–550’ interval.
Class Freq Rel freq Density
0– < 50 8 0.08 .0016
50– < 100 13 0.13 .0026
100– < 150 11 0.11 .0022
150– < 200 21 0.21 .0042
200– < 300 26 0.26 .0026
300– < 400 12 0.12 .0012
400– < 500 4 0.04 .0004
500– < 600 3 0.03 .0003
600– < 900
2 0.02 .00007
100 1.00
z: Count Percent
0 13 27.66
1 11 23.40
2 3 6.38
3 7 14.89
4 5 10.64
5 3 6.38
6 3 6.38
8 2 4.26
N ¼ 47
.894, .830
Class Freq Rel freq
0– < 100 21 0.21
100– < 200 32 0.32
200– < 300 26 0.26
300– < 400 12 0.12
400– < 500 4 0.04
500– < 600 3 0.03
600– < 700 1 0.01
700– < 800 0 0.00
800– < 900
1 0.01
100 1.00
Class Freq Class Freq
10– < 20 8 1.1– < 1.2 2
20– < 30 14 1.2– < 1.3 6
30– < 40 8 1.3– < 1.4 7
40– < 50 4 1.4– < 1.5 9
50– < 60 3 1.5– < 1.6 6
60– < 70 2 1.6– < 1.7 4
70– < 80
1 1.7– < 1.8 5
40 1.8– < 1.9
1
40
Class interval Freq Rel. Freq.
0–< 50 9 0.18
50–< 100 19 0.38
100–< 150 11 0.22
150–< 200 4 0.08
200–< 250 2 0.04
250–< 300 2 0.04
300–< 350 1 0.02
350–< 400 1 0.02
> ¼ 400
1 0.02
50 1.00
Chapter 1
815
b.
The distribution of the natural logs of the original data
is much more symmetric than the original.
c. .56, .14.
29. d. The frequency distribution is:
Class
Relative
frequency Class
Relative
frequency
0 < 150 .193 900 < 1050 .019
150 < 300 .183 1050 < 1200 .029
300 < 450 .251 1200 < 1350 .005
450 < 600 .148 1350 < 1500 .004
600 < 750 .097 1500 < 1650 .001
750 < 900 .066 1650 < 1800 .002
1800 < 1950 .002
The relative frequency distribution is almost unimodal
and exhibits a large positive skew. The typical middle
value is somewhere between 400 and 450, although the
skewness makes it difficult to pinpoint more exactly than
this.
e. .775, .014
f. .211
31. a. 5.24
b. The median, 2, is much lower because of positive
skewness.
c. Trimming the largest and smallest observations yields
the 5.9% trimmed mean, 4.4, which is between the
mean and median.
33. a. A stem-and leaf display:
32 55 Stem: ones
33 49 Leaf: tenths
34
35 6699
36 34469
37 03345
38 9
39 2347
40 23
41
42 4
The display is reasonably symmetric, so the mean and
median will be close.
b. 370.7, 369.50.
c. The largest value (currently 424) could be increased
by any amount without changing the median. It can be
decreased to any value at least 370 without changing
the median.
d. 6.18 min; 6.16 min
35. a. 125.
b. If 127.6 is reported as 130, then the median is 130, a
substantial change. When there is rounding or grouping,
the median can be highly sensitive to a small change.
37. ~x ¼ 92; x
trð25Þ
¼ 95:07; x
trð10Þ
¼ 102:23; x ¼ 119:3
Positive skewness causes the mean to be larger than the
median. Trimming moves the mean closer to the median.
39. a.
y ¼ x þ c,
~
y ¼ ~x þ c
b.
y ¼ c
x,
~
y ¼ c
~
x
41. a. 25.8 b. 49.31 c. 7.02 d. 49.31
43. a. 2887.6, 2888 b. 7060.3
45. 24.36
47. $1,961,160
49. 3.5; 1.3, 1.9, 2.0, 2.3, 2.5
51. a. 1, 6, 5
b. The box plot shows positive skewness. The two
longest runs are extreme outliers.
c. outlier: greater than 13.5 or less than 6.5
extreme outlier: greater than 21 or less than 14
d. The largest observation could be decreased to 6
without affecting f
s
53. a. The mean is 27.82, the median is 26, and the 5%
trimmed mean is 27.38. The mean exceeds the
median, in accord with positive skewness. The
trimmed mean is between the mean and median, as
you would expect.
b. There are two outliers at the high end and one at the
low end, but there are no extreme outliers. Because
the median is in the lower half of the box, the upper
whisker is longer than the lower whisker, and there are
two high outliers compared to just one low outlier, the
plot suggests positive skewness.
55. The two distributions are centered in about the same
place, but one machine is much more variable than the
other. The more precise machine produced one outlier,
but this part would not be an outlier if judged by the
distribution of the other machine.
57. All of the Indian salaries are below the first quartile of
Yankee salaries. There is much more variability in the
Yankee salaries. Neither team has any outliers.
61. The three flow rates yield similar uniformities, but the
values for the 160 flow rate are a little higher.
63. a. 9.59, 59.41. The standard deviations are large, so it is
certainly not true that repeated measurements are
identical.
b. .396, .323. In terms of the coefficient of variation, the
HC emissions are more variable.
65. 10.65
67. a.
y ¼ ax þ b: s
2
y
¼ a
2
s
2
x
:
b. 100.78, .572
Class interval Freq. Rel. Freq.
2.25 < 2.75 2 0.04
2.75 < 3.25 2 0.04
3.25 < 3.75 3 0.06
3.75 < 4.25 8 0.16
4.25 < 4.75 18 0.36
4.75 < 5.25 10 0.20
5.25 < 5.75 4 0.08
5.75 < 6.25 3 0.06
816 Chapter 1
69. The mean is .93 and the standard deviation is .081. The
distribution is fairly symmetric with a central peak, as
shown by the stem and leaf display:
Leaf unit ¼ 0.010
77
811
8 556
9 22333344
955
10 04
10 55
71. a. Mode ¼ .93. It occurs four times in the data set.
b. The Modal Category is the one in which the most
observations occur.
73. The measures that are sensitive to outliers are the mean
and the midrange. The mean is sensitive because all
values are used in computing it. The midrange is the
most sensitive because it uses only the most extreme
values in its computation.
The median, the trimmed mean, and the midfourth are
less sensitive to outliers. The median is the most resistant
to outliers because it uses only the middle value (or
values) in its computation. The midfourth is also quite
resistant because it uses the fourths. The resistance of the
trimmed mean increases with the trimming percentage.
75. a. s
2
y
¼ s
2
x
and s
y
¼ s
x
b. s
2
z
¼ 1 and s
z
¼ 1
77. b. .552, .102 c. 30 d. 19
79. a. There may be a tendency to a repeating pattern.
b. The value .1 gives a much smoother series.
c. The smoothed value depends on all previous values of
the time series, but the coefficient decreases with k.
d. As t gets large, the coefficient (1 – a)
t–1
decreases to
zero, so there is decreasing sensitivity to the initial
value.
Chapter 2
1. a. A\B
0
b. A[B c. (A\B
0
) [ (B\A
0
)
3. a. S ¼ {1324, 1342, 1423, 1432, 2314, 2341, 2413,
2431, 3124, 3142, 4123, 4132, 3214, 3241,
4213, 4231}
b. A ¼ {1324, 1342, 1423, 1432}
c. B ¼ {2314, 2341, 2413, 2431, 3214, 3241, 4213,
4231}
d. A[B ¼ {1324, 1342, 1423, 1432, 2314, 2341, 2413,
2431, 3214, 3241, 4213, 4231}
A\B ¼ ∅
A
0
¼ {2314, 2341, 2413, 2431, 3124, 3142, 4123,
4132, 3214, 3241, 4213, 4231}
5. a. A ¼ { SSF, SFS, FSS }
b. B ¼ { SSS, SSF, SFS, FSS }
c. C ¼ { SSS, SSF, SFS }
d. C
0
¼ { SFF, FSS, FSF, FFS, FFF }
A[C ¼ { SSS, SSF, SFS, FSS }
A\C ¼ { SSF, SFS }
B[C ¼ { SSS, SSF, SFS, FSS }
B\C ¼ { SSS SSF, SFS }
7. a. {111, 112, 113, 121, 122, 123, 131, 132, 133, 211,
212, 213, 221, 222, 223, 231, 232, 233, 311, 312, 313,
321, 322, 323, 331, 332, 333}
b. {111, 222, 333}
c. {123, 132, 213, 231, 312, 321}
d. {111, 113, 131, 133, 311, 313, 331, 333}
9. a. S ¼ {BBBAAAA, BBABAAA, BBAABAA,
BBAAABA, BBAAAAB, BABBAAA, BABABAA,
BABAABA, BABAAAB, BAABBAA, BAABABA,
BAABAAB, BAAABBA, BAAABAB, BAAAABB,
ABBBAAA, ABBABAA, ABBAABA, ABBAAAB,
ABABBAA, ABABABA, ABABAAB, ABAABBA,
ABAABAB, ABAAABB, AABBBAA, AABBABA,
AABBAAB, AABABBA, AABABAB, AABAABB,
AAABBBA, AAABBAB, AAABABB, AAAABBB}
b. {AAAABBB, AAABABB, AAABBAB, AABAABB,
AABABAB}
13. a. .07 b. .30 c. .57
15. a. They are awarded at least one of the first two projects,
.36.
b. They are awarded neither of the first two projects, .64.
c. They are awarded at least one of the projects, .53.
d. They are awarded none of the projects, .47.
e. They are awarded only the third project, .17.
f. Either they fail to get the first two or they are awarded
the third, .75.
17. a. .572 b. .879
19. a. SAS and SPSS are not the only packages.
b. .7 c. .8 d. .2
21. a. .8841 b. .0435
23. a. .10 b. .18, .19 c. .41 d. .59 e. .31 f. .69
25. a. 1/15 b. 6/15 c. 14/15 d. 8/15
27. a. .98 b.
.02 c. .03 d. .24
29. a. 1/9 b. 8/9 c. 2/9
31. a. 20 b. 60 c. 10
33. a. 243 b. 3645, 10
35. .0679
37. .2
39. .0456
41. a. .0839 b. .24975 c. .1998
43. a. 1/15 b. 1/3 c. 2/3
45. a. .447, .5, .2
b. P(A|C) ¼ .4, the fraction of ethnic group C that has
blood type A.
P(C|A) ¼ .447, the fraction of those with blood group
A that are of ethnic group C.
c. .211
47. a. Of those with a Visa card, .5 is the proportion who also
have a Master Card.
b. Of those with a Visa card, .5 is the proportion who do
not have a Master Card.
Chapter 2 817