作業十四 ( week16 習題：7.7, 7.11, 7.12, 7.15, 7.19, 7.21, 7.27)
7.7 a.

1
2
E X
1
)

1
2
E X
2
)
 
2 2


1
4
E X
1
)

3
4
E X
2
)



3
4 4


1
3
E X
1
)

2
3
E X
2
)



2
3 3
 b.

1
4
Var X
1
)

1
4
Var X
2
)
 2  2  2
  
4 4 2

1
16
Var X
1
)

9
16
Var X
2
)

5
 2
8
X
5
 2

1
9
Var X
1
)

4
9
Var X
2
)

9
is most efficient since ( )

( )

( ) c.
Relative efficiency between Y and X :
( )
 
( ) 4
1.25
Relative efficiency between Z and X :
( )

10

( ) 9
1.111
7.11
Calculate the width to estimate the population mean, for a.
90% confidence level, n = 100, variance = 169 width = 2ME = 2

 z

2
 n


=
 
13
100



= 4.277 b.
95% confidence interval, n = 120, standard deviation = 25 width = 2ME = 2

 z

2
 n


=
 
25
120



= 8.9461
7.12
Calculate the LCL and UCL: a. x
 z

2
 n
= 50 1.96


40 b. x
 z

2
 n
= 85 2.58


20
64
225
= ( 40.2 , 59.8)
= (81.56 , 88.44)

c. x
 z

2
 n
= 50 = (506.27 , 513.73)
485
7.15 a. n

25, x

2.90,
 
.45, z
.025

1.96
95% confidence interval: x z


 n
= 2.90  1.96(.45/5) = (2.7236 , 3.0764) b. 2.99 – 2.90 = 0.09 =

Z

/ 2
0 .
45

5
Z

/ 2

2

[ 1
 
( 1 )]

0 .
317

1
Confidence level: 100(1–0.3174) = 68.26%
7.19
Time spent driving to work for n = 20 people s = 9.867 a.
Calculate the standard error s n
 9.867
20
= 2.2063 b.
Find the value of t for the 95% confidence interval t v ,

2
 t
19,.025
= 2.093 c.
Calculate the width for a 95% confidence interval for the population mean width = 2ME = 2


 t

2 s n
 
= 9.2356
7.21
n = 16, sample mean = 47,500 miles, sample standard deviation = 4,200 miles a.
Calculate the margin of error for 95% confidence level to estimate the population mean;
ME
 t v ,

2 s n
= 2.131

4200
16
= 2237.55 b.
90% confidence interval: x
 t v ,

2 s n
=
4200
16
= 45,659.35 miles to 49,340.65 miles

7.27 n

24, x

24.375, s

8.9434, t
23,.005

2.807
a.
99% confidence interval:
24.375
 2.807(8.9434/ 24 ) = 19.2506 pounds up to 29.4994 pounds b.
Narrower since the t-score will be smaller for a 90% confidence interval than for a
99% confidence interval.