490 CHAPTER 11 │ NONPARAMETRIC TESTS
21. The claim is “the daily high temperatures do not occur randomly.”
0:HDaily high temperatures occur randomly.
:
a
H Daily high temperatures do not occur randomly. (claim)
22. The claim is “the scores occur randomly.”
0:H The sequence of exam scores is random. (claim)
:
a
H The sequence of exam scores is not random.
23. Answers will vary.
CHAPTER 11 REVIEW EXERCISE SOLUTIONS
1. (a) The claim is “the median number of customers per day is not more than 650.”
0:H median ≤ 650 (claim); :
a
H median > 650
(b) The critical value is 2.
(c) x = 7
(d) Fail to reject 0.H
(e) There is not enough evidence at the 1% level of significance to reject the bank manager’s
claim that the median number of customers per day is no more than 650.
2. (a) The claim is “median credit score for U.S. adults is at least 710.”
0:H median ≥ 710 (claim); :
a
H median < 710
CHAPTER 11 │ NONPARAMETRIC TESTS 491
3. (a) The claim is “median sentence length for all federal prisoners is 2 years.”
0:H median = 2 (claim); :
a
H median ≠ 2
(b) The critical value is 01.645.z=− .
(c) x = 65
( 0.5) 0.5( ) (65 0.5) 0.5(174) 21.5 3.26
6.595
174
xn
zn
+− +− −
== ≈≈−
4. (a) The claim is “there was no reduction in diastolic blood pressure.”
0:H There is no reduction in diastolic blood pressure. (claim)
5. (a) The claim is “there was no reduction in diastolic blood pressure.”
0:H There is no reduction in diastolic blood pressure. (claim)
6. (a) The claim is “the median salary of lawyers 9 months after graduation from law school is
$68,500.
(b) The critical value is 01.96z=− .
(c) x = 49
( 0.5) 0.5( ) (49 0.5) 0.5(125) 13 2.33
5.59
125
xn
zn
+− +− −
== ≈≈−
492 CHAPTER 11 │ NONPARAMETRIC TESTS
7. (a) Independent; Wilcoxon rank sum test
(b) The claim is “there is a difference in the total times required to earn a doctorate degree by
female and male graduate students.”
0:H There is no difference in the total times to earn a doctorate degree by female
and male graduate students.
:
a
H There is a difference in the total times to earn a doctorate degree by female
and male graduate students.
(c) The critical values are 02.575.z±
(d)
11 2
(1)
12(12121) 150
22
R
nn n
µ
++ ++
===
12 1 2
(1)
(12)(12)(12 12 1) 17.321
12 12
R
nn n n
σ
++ ++
=≈
Ordered data Sample Rank
7 F 1.5
7 F 1.5
8 M 4.5
10 M 13
10 M 13
10 M 13
11 F 17.5
11 M 17.5
11 M 17.5
11 M 17.5
12 F 20.5
CHAPTER 11 │ NONPARAMETRIC TESTS 493
8. (a) Dependent; Wilcoxon signed-rant test
(b) The claim is “a new drug affects the number of headache hours experienced by headache
sufferers.”
0:H The new drug does not affect the number of headache hours experienced.
:
a
H The new drug does affect the number of headache hours experienced. (claim)
(c) The critical value is 4.
(d)
Before After Difference Absolute value Rank Signed rank
0.9 1.4 −0.5 0.5 1.5
−1.5
2.3 1.5 0.8 0.8 4 4
3.1 1.9 1.2 1.2 5 5
Sum of negative ranks = −1.5
Sum of positive ranks = 34.5
1.5
s
w=
(e) Reject
0.H
(f) There is enough evidence at the 5% level of significance to support the claim that the new
drug does affect the number of headache hours experienced.
494 CHAPTER 11 │ NONPARAMETRIC TESTS
(c)
123
191.5, 126, 243.5RRR===
2
22
3
12
123
22 2
12 3( 1)
(1)
(191.5) (126) (243.5)12 3(33 1)
33(33 1) 11 11 11
6.741
R
RR
HN
NN n n n
⎛⎞
=++−+
⎜⎟
+⎝⎠
⎛⎞
=++−+
⎜⎟
+⎝⎠
≈
Ordered data Sample Rank
29 L 1.5
29 P 1.5
31 L 12.5
31 P 12.5
31 P 12.5
33 P 25
33 S 25
33 S 25
34 L 28
34 L 28
CHAPTER 11 │ NONPARAMETRIC TESTS 495
10. (a) The claim is “the distributions of the starting salaries in these four fields of engineering are
different.”
0:H There is no difference in the starting salaries among the fields of engineering.
:
a
H There is a difference in the starting salaries among the fields of engineering. (claim)
(b) The critical value is 7.815.
(c)
Ordered data Sample Rank
56.1 Mech 1
56.8 Elec 2
57.1 Mech 3
59 Mech 11.5
59.3 Comp 14
59.3 Comp 14
59.3 Elec 14
59.5 Elec 16
59.7 Elec 17
59.8 Elec 18
59.9 Comp 19
62 Mech 27
62.3 Chem 28
62.4 Comp 29
63.1 Comp 30
63.7 Chem 31
63.9 Chem 32
65.5 Chem 33.5
496 CHAPTER 11 │ NONPARAMETRIC TESTS
1234
349.5, 217.5, 143.5, 109.5RR RR====
2
22 2
3
12 4
1234
12 3( 1)
(1)
R
RR R
HN
NN n n n n
⎛⎞
= +++ − +
⎜⎟
+⎝⎠
11. (a) The claim is “there is a correlation between overall score and price.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
(b) The critical value is 0.786.
(c)
Score Rank Price Rank d d2
93 7 500 6.5 0.5 0.25
91 6 300 5 1 1
90 5 500 6.5
−1.5 2.25
87 4 150 2 2 4
85 2 250 4 −1 1
74 3 200 3 −1 1
69 1 130 1 0 0
29.5d=
∑
29.5d=
∑
ρ
ρ
CHAPTER 11 │ NONPARAMETRIC TESTS 497
(c)
Score Rank Price Rank d d2
86 9 33 7 2 4
84 8 32 6 2 4
82 7 20 3 4 16
81 6 45 9 −3 9
13. (a) The claim is “the stops were not random by gender.”
0:H The traffic stops were random by gender.
:
a
H The traffic stops were not random by gender. (claim)
(b)
1
n = number of F’s = 12
2
n = number of M’s = 13
lower critical value = 8
upper critical value = 19
(c) G = 14 runs
(d) Fail to reject 0.H
(e) There is not enough evidence at the 5% level of significance support the claim that the traffic
stops were not random by gender.
498 CHAPTER 11 │ NONPARAMETRIC TESTS
CHAPTER 11 QUIZ SOLUTIONS
1. (a) The claim is “there is a difference in the hourly earnings of union and nonunion workers in
state and local governments.”
0:H There is no difference in the hourly earnings.
:
a
H There is a difference in the hourly earnings. (claim)
(b) Wilcoxon rank sum test
(c) The critical values are 01.645z±.
(d)
R = sum ranks of nonunion workers = 61
11 2
(1)
10(10 10 1) 105
22
R
nn n
µ
++ ++
===
12 1 2
(1)
(10)(10)(10 10 1) 13.229
12 12
R
nn n n
σ
++ ++
=≈
Ordered data Sample Rank
19.10 N 1
19.85 N 2
24.20 U 9
24.80 U 10.5
24.80 N 10.5
25.05 U 12
25.30 U 13
25.60 U 14.5
32.97 U 20
CHAPTER 11 │ NONPARAMETRIC TESTS 499
2. (a) The claim is “the median number of annual volunteer hours is 52.”
0:H median is 52 (claim); :
a
H median ≠ 52
(b) Sign test
(c) The critical values are 1.96±.
(d) 23x=
()
()()
0.5 0.5
2
23 0.5 0.5 70 2.75
70
2
xn
zn
+−
=
+−
==−
500 CHAPTER 11 │ NONPARAMETRIC TESTS
(d)
1234
223, 97, 44, 164RRRR====
2
22 2
3
12 4
1234
222
12 3( 1)
(1)
(223) (97) (44) (164)12 3(32 1)
32(32 1) 8 8 8 8
25.957
R
RR R
HN
NN n n n n
⎛⎞
=+++−+
⎜⎟
+⎝⎠
⎛⎞
=+++−+
⎜⎟
+⎝⎠
≈
Ordered data Sample Rank
149.8 S 1
150.9 S 2
161 S 3
185.3 MW 10
187.1 MW 11
188.9 MW 12
189.9 W 13
190.5 S 14
191.9 MW 15
200.9 MW 16
201.9 W 17
205.1 MW 18
206.3 W 19
218.5 W 20
220 W 21
270.2 NE 32
CHAPTER 11 │ NONPARAMETRIC TESTS 501
4. (a) The claim is “days with rain are not random.”
0:H The days with rain are random.
:
a
H The rains with rain are not random. (claim)
(b) Runs test
5. (a) 0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
(b) Spearman rank correlation coefficient.
(c) The critical value is 0.829.
Larceny Rank Motor vehicle Rank d d2
1403 1 161 1 0 0
1506 2 608 2 0 0
24d=
∑
2
22
66(4)
1 1 0.886
(1) 6(61)
c
d
rnn
=− =− ≈
−−
∑
(e) Reject
0.H
(f) There is enough evidence at the 10% level of significance to conclude that there is a
correlation between the number of larceny-thefts and the number of motor vehicle thefts.
CUMULATIVE REVIEW FOR CHAPTERS 9−11
1. (a) r ≈ 0.815
There is a strong positive linear correlation.
502 CHAPTER 11 │ NONPARAMETRIC TESTS
(b)
0:H 0
ρ
=
:
a
H 0
ρ
≠ (claim)
(c)
ˆ1.264 1.581yx=−
2. The claim is “there is a difference in the weekly earnings of workers who are union members and
workers who are not union members.”
0:H Median (Union) = Median (Nonunion)
:
a
H Median (Union) ≠ Median (Nonunion) (claim)
The critical values are 01.96.z=±
Ordered data Sample Rank
803 N 8
814 N 9
855 U 10
862 N 11
884 U 12
904 U 13
930 U 14
CHAPTER 11 │ NONPARAMETRIC TESTS 503
R = sum ranks of nonunion workers = 44
11 2
(1)
8(8 9 1) 72
R
nn n
µ
++ ++
===
3. The claim is “the median age of people with mutual funds is 50 years.”
0:H Median = 50 (claim)
:
a
H Median ≠ 50
The critical value is 3.
x = 6
Fail to reject 0.H There is not enough evidence at the 1% level of significance to reject the
company’s claim that the median age of people with mutual funds is 50 years.
5. (a) ˆ11,182 174.53(91) 104.41(88) 17,876.15y=+−= pounds per ounce.
(b)
ˆ11,182 174.53(110) 104.41(98) 20,148.12y=+ − = pounds per ounce.
6. The claim is “the standard deviations reading test scores for eigth grade students are the same in
Colorado and Utah.”
0:H 22
12
σσ
= (claim)
:
a
H 22
12
σσ
≠
02.46F=
504 CHAPTER 11 │ NONPARAMETRIC TESTS
7. The claim is “the distributions of the annual household incomes in these regions are different.”
0:H The medians are all equal.
:
a
H The medians are not all equal. (claim)
The critical value is 11.345.
1234
126, 91, 37, 152RRRR====
2
22 2
3
12 4
1234
12 3( 1)
(1)
R
RR R
HN
NN n n n n
⎛⎞
= +++ − +
⎜⎟
+⎝⎠
Ordered data Sample Rank
41.5 S 1
44.4 S 2
49.2 S 9.5
49.2 S 9.5
49.3 MW 11
50.0 NE 12
50.7 MW 13
51.4 W 14
51.6 NE 15
51.8 MW 16
52.0 MW 17
8. Claimed distributions:
Response Distribution
None 5%
Little 16%
Half 31%
Most 33%
All 15%
0:H The distribution is as claimed. (claim)
2
09, 488
χ
=
8.430
χ
≈
Fail to reject 0.H There is not enough evidence at the 5% level of significance to conclude that
the distribution of how much parents intend to contribute to their children’s college costs differs
from the claimed or expected distribution.
9. (a) 20.733r≈
Metacarpal bone length explains about 73.3% of the variability in height. About 26.7% of the
variation is unexplained.
(b) 4.255
e
s≈
The standard error of estimate of the height for a specific metacarpal bone length is about
506 CHAPTER 11 │ NONPARAMETRIC TESTS
10. The claim is “there is a correlation between overall score and the price.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
The critical value is 0.643.
Score Rank Price Rank d d2
74 3.5 77 3.5 0 0
82 7 96 6 1 1
78 5 77 3.5 1.5 2.25
84 8 116 8 0 0