Chapter 16
16.1 (a)
2
12 12
212
12 21
2( )
/2 ()()
24
xx xx
x
txx
sxx xx
16.2 When
Tk
then
(1)
2
nn
Tk
and then
(1)
() 2
(1)
= 2
nn
PT k P T k
nn
PT k
16.3
(1) (1)
2
22
nn nn
TT T T T X
(1) (1)
() 2 0
42
nn nn
EX
by Theorem 16.1
(1)(21)
var( ) 4 by Theorem 16.1
24
(1)(21)
6
nn n
X
nn n
246 Mathematical Statistics, 8E
16.6
i
r
——-———|—————
——-———|—————
12
1
i
nn r
16.7
11 1212 11
11 2
(1)( )( 1) (1)
22 2
nn n n n n nn
UW W
16.8 Ranking of x‘s are
1
123
…
n
rr r r
1
r
2
r
3
r
1
n
r
y‘s ——-–|——-——-|—-—-——–|–——-—-—|–————–|—-
1
1r
21
1rr
32
1rr
…
11
1
1
nn
rr
Chapter 16 247
16.9
2
1
1
12 1
(1) 2
k
i
i
i
Rn
Hn
nn n
1
i
i
16.10
..
(1)
, , 2
ii i
nn
TR nnT
2
(1)(21)
6
nn n
x
22
(1)(21)1(1) ( 1)
(d.f. 1)
6212
nn n nn nn
SST n
n
248 Mathematical Statistics, 8E
16.11 k + 1 runs of first kind and k
runs of second kind in
12
11
1
nn
kk
ways
16.12
12
7, 3nn
62 62 62
200 10 01
21 84
(2) ; (3)
10 120 60 120 120 60
7
ff
16.13
54
233 2104 80
(8) 11 462 462
6
f
Chapter 16 249
16.14
77
00 2
(2) 0.000155
16 12,870
8
f
77 77
2
10 01 14
(3) 0.001088
12,870 12,870
f
16.15 W = 0 makes
(1)
2
i
kn
R
for each value of i; it reflects a complete lack of association.
There is complete agreement, for instance, when
i
R
ki
and
16.16
1
20 10
2
μ
11
20 2.236
22
σ
250 Mathematical Statistics, 8E
16.17 Differences are
1.3 0.9 1.1 3.8 3.1 2.6 1.8 2.5 1.2 2.4
− + − − + − − − − −
11 7.5 9 20 19 17 13 16 10 15
16.18 There are x = 12 plus signs among n = 16
0.05
α
p = 0.5 against p > 0.50, p–value
( 12) 0.0381px
Since p-value is less than 0.05, reject the null hypothesis.
16.19 1.15 0.85 4.75 −0.37 2.09 6.63 −2.35 0.27
8 6 14 4 11 16 12 3
16.20 n = 10,
0.05
α
(a) based on T; reject if
0.05 8TT
8T
16.21 n = 10,
0.01
α
(a) based on T; reject if
0.01 3TT
16.22
035
μ
against
35, 0.05
μ
α
, n = 11
3 8 1 −6 9 −7 5 15 4 12 −2
3 8 1 6 9 7 5 11 4 10 2
Chapter 16 251
16.23 15 18 20 22 25 27 28 29 32 35 36 38
2 2 2 2 1 2 1 2 1 1 1 1
μ
16.24
0.05
α
12
μ
μ
10.01
10UU
1813.552 7 26.5W
15.5 10U
16.25
0.05
α
12
μ
μ
1 2 0.05
10, 12, 49nnUU
16.26
2
15 12 15 12 28 88 90
90, 420, 20.5, 0.10
2 12 20.5
z
μσ σ
Since
0.10z
falls between
1.96 and 1.96
, null hypothesis cannot be rejected.
16.28 B B B B A B A B A A A A
000012 3U
16.29
12
14, 8, 5, 0.05nnu
α
0.025 6u
Since
56u
, null hypothesis of randomness must be rejected.
252 Mathematical Statistics, 8E
16.32
12
38, 22, 28, 0.05nn u
α
23822 1 28.87
60
μ
16.33
12
24, 24, 30, 0.01nnu
α
16.35 Median is 30.5 and we get
b b a b b b a a a b a a a b a b a b b b a a a b
16.36 Median is 99.7
b a a b a a a b b b a b a b a b a b a b a b a b
12
12, 12, 19, 0.05nn uc
21212 113
μ
16.37 11.3 12.2 13.0 13.2 14.1 14.7 14.9 15.2 15.3 15.4
A B A A A B A B B A
16.2 16.6 16.9 17.0 18.3 18.9 19.4 19.8 21.2
B A A A B B B B B
Chapter 16 253
16.38
x
R
y
R
d
2137d
13 12 1
14 11 3
1 2 −1
16.5 14.5 2
2.5 1 1.5
6(137)
1 1 0.14 0.86
18.323
s
r
15 16 −1
16.39
0.86 0
1/ 18 1
z
= 0.86(4.423) = 3.55
Since 3.55 exceeds 1.96, the value of
s
r
is significant.
16.42
12 345 6 78
910
15, 12, 7, 15, 29, 10, 11, 25,
(1)311
25, 15, 16.5
22
RR RR R R RR
kn
RR
254 Mathematical Statistics, 8E
16.43 A and B
2
86d
6(86)
1 1 0.521 0.479
s
r
16.44 Number of plus signs = 25 out of n = 36
0.01
α
24.5 18
35(0.5) 18, 36(0.5)(0.5) 3, 2.16
3
z
μσ
using continuity correction.
Since 2.16 is less than
0.01
z
= 2.33, null hypothesis cannot be rejected.
16.45 0.1 1.1 0.3 1.1 1.0 0.7 0.6 0.4 0.8 1.0
− + + + + + + + + −
3.5 30.5 3.5 3.5 23 8.5 17 8.5 3.5 17 27.5
146
T
2
36 37 36 37 73 146 333
333, 4051.5, 63.65, 2.94
4 24 63.65
z
μσ σ
0.01
α
Since
2.94 2.33
, null hypothesis must be rejected.
16.47 43 35 13 11 6 18 12 6 2 7 3 10
Chapter 16 255
16.48 Number of plus signs x = 7 n = 24
0.05
α
24(0.5) 12 and = 24(0.5)(0.5)
μσ
= 2.45
16.49 −5 −13 −6 −7 9 −8 −1 6 −7 7 −11
9 24 12 15 20 18 1.5 12 15 15 21
16.50 −5 9.4 11.1 −9.3 −1.5 15.6 29 4.3 12.9 −0.9
11 16 17 15 4 22 24 9 19 2
13 7.7 11.2 −0.1 3.8 −1.9 26.3 5.5 15.4
20 14 18 1 7 6 23 12 21
3.9 1.6 6.2 4.7 −1.4
8 5 13 10 3
16.51 5 −12 −3 8 11 −8 −16 13
7.5 16 4 11.5 15 11.5 19 17
256 Mathematical Statistics, 8E
(a)
7.5 11.5 15 17 4 7.5 1 13 10 9 6 101.5
T
19 20 101.5 98.5
2
T
, T = 98.5
0.05
45
T
16.52
0.05
α
12
μ
μ
12
20
nn
16.53
0.05
α
12
μ
μ
12
16
nn
1
307
W
2
1
16 17 16 16 16 16 33
307 171, = 128, 704
U
μσ
16.54
0.05
α
2
0.05,3
7.815
χ
1
4 7 10 14 18 53
R
2
512151620 68
R
16.55
123
10
nnn
0.05
α
d.f. = 2
2
0.05,2
5.991
χ
1
1.5 5 7.5 10.5 12 13 15.5 18 25 28 136
R
Chapter 16 257
16.56
12 3
8, 10, 8
nn n
0.01
α
2
0.01,2
9.210
χ
1
3 6 12 13 15 21 25 26 121
R
16.57 Median = 21.5
b b b b b bab b b b b bbaabbbbbbbaaab b b a a b aaaa
a b a a a a a a a a a a a a
16.58 Median is 5
16.59 Median = 138
0.05
α
b b b b b a a b b a a a a b a b b b a a a a b b b a b b a a a a
12
16, 16, 12nn u
Since
1.80z
is less than
1.645
; the null hypothesis of randomness must be rejected; there
seems to be a trend.