CHAPTER 11 │ NONPARAMETRIC TESTS 473
(c)
Ordered data Sample Rank
25.5 KT 1
27.1 WV 2
27.8 SC 3
28.9 WV 4
34.2 KT 12
35.4 SC 13
36.6 NC 14
37.4 WV 15
38.2 WV 16
38.9 NC 17
40.5 NC 18
40.9 SC 19
41.5 SC 20
42.6 WV 21
43.1 KT 22
51.3 NC 23
54.7 KT 24
1234
72.5, 89.5, 75, 63RRRR====
2
22 2
3
12 4
1234
12 3( 1)
(1)
R
RR R
HN
NN n n n n
⎛⎞
= +++ − +
⎜⎟
+⎝⎠
474 CHAPTER 11 │ NONPARAMETRIC TESTS
(c)
95 Soft drinks 10
96 Soft drinks 11
100 Teas 12
106 Teas 13
141 Energy drinks 14
1234
105, 56, 104, 35RRRR====
2
22 2
3
12 4
1234
22 22
12 3( 1)
(1)
(105) (56) (104) (35)12 3(24 1)
24(24 1) 5 7 6 6
R
RR R
HN
NN n n n n
⎛⎞
= +++ − +
⎜⎟
+⎝⎠
⎛⎞
=+++−+
⎜⎟
+⎝⎠
CHAPTER 11 │ NONPARAMETRIC TESTS 475
7. Kruskal-Wallis results:
Data stored in separate columns.
Chi Square = 8.0965185 (adjusted for ties)
DF = 2
P-value = 0.0175
Column N Median Ave. Rank
A 6 5 6.75
B 6 8.5 14.5
C 6 5 7.25
8. Kruskal-Wallis results:
Data stored in separate columns.
B 6 8.5 20.5
C 6 5 12.333333
D 6 3 5.0833335
P ≈ 0.0022 < 0.01, so reject 0.H There is enough evidence at the 1% level of significance to
conclude that the distributions of the number of job offers at all four colleges are different.
476 CHAPTER 11 │ NONPARAMETRIC TESTS
Ordered data Sample Rank
1 NE 2.5
1 MW 2.5
1 S 2.5
4 MW 13.5
4 W 13.5
5 NE 19
5 MW 19
5 S 19
5 S 19
5 S 19
5 W 19
5 W 19
6 NE 26
1 234
220.5, 171.5, 159.5, 151.5RRRR====
2
22 2
3
12 4
1234
12 3( 1)
(1)
R
RR R
HN
NN n n n n
⎛⎞
= +++ − +
⎜⎟
+⎝⎠
CHAPTER 11 │ NONPARAMETRIC TESTS 477
(b)
Variation
Sum of
squares
Degrees of
freedom
Mean
squares
F
Between 9.17 3 3.06 0.52
Within 194.72 33 5.90
Because 0.01
α
= and the test is two–tailed, use the 10.005
α
= table. The critical value is
478 CHAPTER 11 │ NONPARAMETRIC TESTS
Ordered data Sample Rank
28 W 1
32 W 2
70 W 9.5
72 NE 11.5
72 S 11.5
74 S 13.5
74 W 13.5
108 NE 25
113 MW 26
118 W 27
120 MW 28
129 MW 29
123 4
253, 307, 113.5, 106.5RR R R=== =
2
22 2
3
12 4
1234
2222
12 3( 1)
(1)
(253) (307) (113.5) (106.5)12 3(39 1)
39(39 1) 10 11 8 10
R
RR R
HN
NN n n n n
⎛⎞
= +++ − +
⎜⎟
+⎝⎠
⎛⎞
=+++−+
⎜⎟
+⎝⎠
CHAPTER 11 │ NONPARAMETRIC TESTS 479
Reject 0.H There is enough evidence at the 1% level of significance to support the claim that
the mean energy consumptions are different.
(b)
Variation
Sum of
squares
Degrees of
freedom
Mean
squares
F
11.4 RANK CORRELATION
11.4 Try It Yourself Solutions
1a. The claim is “there is a correlation between the number of males and females who receive
doctoral degrees.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
b. 0.01
α
=
c. The critical value is 0.929.
d.
Male Rank Female Rank d d2
25 3.5 20 1.5 2 4
24 1.5 20 1.5 0 0
26.5d=
∑
e.
2
22
66(6.5)
1 1 0.884
(1) 7(71)
c
d
rnn
≈− =− ≈
−−
∑
480 CHAPTER 11 │ NONPARAMETRIC TESTS
11.4 EXERCISE SOLUTIONS
1. The Spearman rank correlation coefficient can be used to describe the relationship between linear
and nonlinear data. Also, it can be used for data at the ordinal level and it is easier to calculate by
2. Both the Spearman rank correlation coefficient and the Pearson correlation coefficient range from
−1 to 1, inclusive.
3. The ranks of corresponding data pairs are identical when s
r is equal to 1.
4. The value of s
r represents the correlation coefficient of the sample data, whereas s
ρ
represents
the correlation coefficient of the entire population.
5. (a) The claim is “there is a correlation between debt and income in the farming business.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
(b) The critical value is 0.929.
(c)
Debt Rank Income Rank d d2
19,955 7 28,926 7 0 0
10.480 4 8,630 2 2 4
28d=
∑
2
2
6
1 0.857
(1)
c
d
rnn
≈− ≈
−
∑
(d) Fail to reject 0.H
ρ
ρ
CHAPTER 11 │ NONPARAMETRIC TESTS 481
(c)
Score Rank Price Rank d d2
85 11 2600 9 2 4
78 10 2800 10 0 0
77 9 3700 11
−2 4
259.5d=
∑
2
2
6
1 0.730
(1)
c
d
rnn
≈− ≈
−
∑
(d) Reject
0.H
(e) There is enough evidence at the 5% level of significance to conclude that there is a
correlation between the overall score and price.
7. (a) The claim is “there is a correlation between wheat and oat prices.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
(b) The critical value is 0.833.
(c)
Oat Rank Wheat Rank d d2
1.10 1 2.62 1 0 0
1.59 4 2.78 2 2 4
1.81 6 3.56 6 0 0
26d=
∑
2
2
6
1 0.950
(1)
c
d
rnn
≈− ≈
−
∑
8. (a) The claim is “there is a correlation between the overall score and the price.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
(b) The critical value is 0.497.
(c)
Score Rank Price Rank d d2
73 12 230 5 7 49
65 7 400 9
−2 4
60 3.5 600 10.5
−7 49
2330d=
∑
2
2
6
1 0.154
(1)
c
d
rnn
≈− ≈−
−
∑
9. The claim is “there is a correlation between science achievement scores and GNI.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
The critical value is 0.700.
Science average Rank GNI Rank d d2
534 9 1307 3 6 36
495 5 2467 6
−1 1
516 7 3207 7 0 0
CHAPTER 11 │ NONPARAMETRIC TESTS 483
10. The claim is “there is a correlation between mathematics achievement scores and GNI.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
The critical value is 0.700.
Math average Rank GNI Rank d d2
527 9 1307 3 6 36
496 5 2467 6
−1 1
504 7 3207 7 0 0
462 2 1988 5
−3 9
523 8 4829 8 0 0
2108d=
∑
2
2
6
1 0.100
(1)
c
d
rnn
≈− ≈
−
∑
Fail to reject 0.H There is not enough evidence at the 5% level of significance to conclude that
there is a correlation between mathematics achievement scores and GNI.
11. The claim is “there is a correlation between science and mathematics achievement scores.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
The critical value is 0.700.
Science average Rank Science average Rank d d2
534 9 527 9 0 0
495 5 496 5 0 0
484 CHAPTER 11 │ NONPARAMETRIC TESTS
13. The claim is “there is a correlation between average hours worked and the number of on-the-job
injuries.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
The critical values are 1.96 0.346.
1331
z
n
±±
=≈±
−−
Hours worked Rank Injuries Rank D D2
47.6 31 16 2 29 841
44.1 11 33 31.5
−20.5 420.25
45.6 24 25 17.5 6.5 42.25
42.3 2 32 30
−28 784
45.5 21.5 26 21 0.5 0.25
41.8 1 28 26
−25 625
43.1 5.5 22 12
−6.5 42.25
44.4 12 19 5 7 49
44.5 14 23 13.5 0.5 0.25
43.7 9.5 20 7 2.5 6.25
44.9 19 28 26
−7 49
47.8 32 24 15.5 16.5 272.25
∑
26673d=
∑
2
2
6
1 0.115
(1)
c
d
rnn
≈− ≈−
−
∑
CHAPTER 11 │ NONPARAMETRIC TESTS 485
Fail to reject 0.H There is not enough evidence at the 5% level of significance to conclude that
there is a correlation between average hours worked and the number of on-the-job injuries.
14. The claim is “there is a correlation between average hours worked and the number of on-the-job
injuries.”
0: 0
s
H
ρ
=; :
a
H 0
s
ρ
≠ (claim)
The critical values are 1.96 0.341.
1341
z
n
±±
=≈±
−−
Hours worked Rank Injuries Rank D D2
40.5 26 12 6 20 400
38.3 14 13 10.5 3.5 12.25
37.8 8 19 26.5
−18.5 342.25
38.2 12 18 23.5
−11.5 132.25
38.6 15 22 28.5
−13.5 182.25
36.2 4 15 17.5
−13.5 182.25
38.7 16 18 23.5
−7.5 56.25
36 2 11 4
−2 4
486 CHAPTER 11 │ NONPARAMETRIC TESTS
11.5 THE RUNS TEST
11.5 Try It Yourself Solutions
1a. PPP F P F PPPP FF P F PP FFF PPP F PPP
a
b. 0.05
α
=
c. M FFF MM FF M F MM FFF
1
n = number of F’s = 9
2
n = number of M’s = 6
:
a
b. 0.05
α
=
c. 1
n = number of N’s = 21
2
n = number of S’s = 10
G = number of runs = 17
CHAPTER 11 │ NONPARAMETRIC TESTS 487
17 14.55 1.03
2.38
G
G
G
z
µ
σ
−−
=≈ ≈
11.5 EXERCISE SOLUTIONS
1. Answers will vary. Sample answer: It is called the runs test because it considers the number of
runs of data in a sample to determine whether the sequence of data was randomly selected.
7. 1
n = number of T’s = 6 8. 1
n = number of U’s = 8
2
n = number of F’s = 6 2
n = number of D’s = 6
9. 1
n = number of M’s = 10 10. 1
n = number of A’s = 13
2
n = number of F’s = 10 2
n = number of B’s = 9
11. 1
n = number of T’s = 6 12. 1
n = number of M’s = 9
1
n = number of F’s = 6 2
n = number of F’s = 3
too high: 11; too low: 3 too high: 8; too low: 2
488 CHAPTER 11 │ NONPARAMETRIC TESTS
(d) Fail to reject 0.H
(e) There is not enough evidence at the 5% level of significance to support the claim that the coin
tosses were not random.
16. (a) The claim is “the sequence of majority parties is not random.”
0:H The sequence of majority parties is random.
:
a
H The sequence of majority parties is not random. (claim)
G
(d) Reject
0.H
(e) There is enough evidence at the 5% level of significance to conclude that the sequence is not
random.
17. (a) The claim is “the sequence of World Series winning teams is not random.”
0:H The sequence of leagues of winning teams is random.
:
a
H The sequence of leagues of winning teams is not random. (claim)
(b)
1
n = number of N’s = 17
2
n = number of A’s = 23
18. (a) The claim is “the digits were not randomly generated.”
0:H The sequence of digits was randomly generated.
:
a
H The sequence of digits was not randomly generated. (claim)
CHAPTER 11 │ NONPARAMETRIC TESTS 489
(b)
1
n = number of O’s = 16
2
n = number of E’s = 16
lower critical value = 11; upper critical value = 23
(c) G = 9 runs
(d) Reject
0.H
(e) There is enough evidence at the 5% level of significance to support the claim that the
sequence of digits was not randomly generated.
20. (a) The claim is “the sequence of past winners is not random.”
0:H The sequence of past winners is random.
:
a
H The sequence of past winners is not random. (claim)
(b)
1
n = number of F’s = 42
2
n = number of A’s = 16
01.96z=±
(c) G = 32 runs