Chapter 11
11.2 (a)
12
12
12
[0 ( )] 1
() 1
()
pkxx
px x k
px x k
θα
θα
θα
2
22
11
2k
θα
θ
2
1
2k
α
2
1
2
k
α
1
2
k
α
Chapter 11 163
11.3
()1
1
pR cR
pR
c
θα
θθα
11.4 By inspection
/
3/2 /22/3
/2 /3 2 /3
2
zz zz
zzz
αα α α
αα α
length of first confidence interval is less than that of 2nd confidence interval
11.5 Length of confidence interval:
Since the normal density function
()
fx
is decreasing for
21
0, x
δδ
, thus
/2
2
k
Lz n
α
σ
164 Mathematical Statistics, 8E
11.6
/2
1px z n
α
σ
μ
α
11.7 Substitute
/2, 1 /2
for
n
s
tz
nn
αα
σ
If
x
, the mean of a random sample of size n from a formal population with the mean
μ
, is
used as an estimate of
μ
, we can assert with
(1 )100%
α
confidence that the error is less than
/2, 1
n
s
tn
α
.
μ
μ
12
11.9
22222
1212
12 12 12
11 2
() 22 2
p
nnnn
ES nn nn nn
σσ σσ
therefore unbiased
11.10
12 2 1
2
12
12 1 2 12
()( ) 1
11 (2)
22
p
Zxx
TYnn S
nn n n nn
μ
μ
σ
Chapter 11 165
11.13
/2
;
(1 )
xn
zn
α
θ
θθ
/
2
(1 )
xn z
n
α
θ
θθ
Let
*
θ
= value of
θ
with
θθθ
closest to
1
2
. By Theorem 11.7,
/2
*(1 *)
ez n
α
θθ
and
2
/
2
2
*(1 *) z
ne
α
θθ
11.16 If
12
nnn
, then
112 2
/2
ˆˆˆˆ
(1 ) (1 )
Ez n
α
θθθθ
The right-hand side of this inequality is maximized when
12
1.
2
θθ
Thus,
/2
1
2
Ez n
α
,
2
2
/
2
,
2
z
En
α
and
2
/
2
2
2
z
nE
α
.
/
/
166 Mathematical Statistics, 8E
11.20 n = 150 9.4
σ
9.4 1.96(9.4)
1.96 1.50
12.247
150
E
11.21 9.4
61.8 2.575 61.8 1.98, 59.82 63.78
150
μ
11.24
0.005
;
s
xz n
45
52.80 2.575 ,
64
or (51.35, 54.25).
11.25
0.025
2.68
1.96 0.83 min.
40
s
ez n
11.26
0.025
9.4 900 150
1.96 1.37.
1 900 1
150
Nn
ez N
n
σ
Chapter 11 167
11.33
22
12
12 0.05
12
() ;xx z nn
σσ
22
4.8 3.5
5.2 1.645 16 25
, or ( 7.49, 2.91).
11.34
22
12
12 0.05
12
() ;
ss
xx z nn
22
19.4 18.8
7.4 2.575 61
, or ( 16.31,1.51).
11.36
11 2 2
8260, 251.89, 7930, s 206.52xs x
22
2
4(251.89) 4(206.52) 53,049.54
8
p
s
230.32
p
s
11
8260 7930 3.355(230.32) 55
12
330 488.75
158.75 818.75 million calorie per ton
μμ
168 Mathematical Statistics, 8E
11.41 (0.76)(0.24)
1.96 0.053
250
e
11.42 (0.18)(0.82)
0.18 2.575 100
0.18 0.099
0.081 0.279
θ
11.45
2
2
(1.96) 2401
4(0.02)
n
11.46
2
1.96
(0.03)(0.70) (0.21)(9604) 2017
0.02
n
Chapter 11 169
11.49 84 0.336
250 156 0.624
250
(0.336)(0.664) (0.624)(0.376)
(0.336 0.624) 1.96 250 250
0.288 0.084
12
0.372 0.204
θθ
11.51 (0.096)(0.904) (0.170)(0.830)
2.33 500 400
2.33(0.022939) 0.053
e
11.54
22
2
11(0.625) 11(0.625)
19.675 4.575
σ
2
0.2184 0.939
σ
0.47 0.97
σ
11.55 4.5 4.5
2.575 2.575
11
128 128
σ
3.67 5.83
σ
170 Mathematical Statistics, 8E
2
6.39
σ
2
σ
11.60 Using MINITAB we enter the data into C1 and we give the command
MTB> Tinterval 95.0 C1
Obtaining
N MEAN STDEV SEMEAN 95.0 PERCENT C.I.
20 6.145 1.467 0.328 (5.458, 6.832)