7
1.

= 99 = 7 = 99 (a) 7 =p =7 2 x x

4

(b)

x

x

= 99 = p7 = 7 5
25

3.

n = 5000

X

= 170

=5
165 ; 170 5

(a) P (165 < X

< 175) = P

2:99)

8

P X > 25
5= 2

)P

Z>

25 ; 23 p 5= n

9

:

<

=

t:

>

= 0:0082
N (0:05 0:01) s 0:075 ; 0:05 0:05 (1 ; 0:05)
475
A

pn = 2:4 ) n = 36
0:05 = 23:75 > 56 0:03 ; 0:05 475
? @

7. np = 475

F

z1 =

s

0:05

(1 ; 0:05)

= ;2

z2 =

= ;2:5
#

P (0:03 < F < 0:075) = P (;2 < Z < 2:5) = 0:971
9. (a) df

= 18 (b) df = 29

(18) = 34:8053 2 0 975 (29) = 16:0471
2 0:01
:

2

7
(c) df

) = 1 ; = 0:99 2 = 0 01 (4) = 12:2767 2 < (n ; 1)K =P 11. P (S 2 < K ) = P 2
2

P(

=4

<

2

:

2

<

11K 25

= 0:95

)

11K 25
B

= 19:6751 ) K = 44:7161

1. C
1. F 3. K

D G L
:

E H M I N

7.2 J
1 O

> 6 P Q Q L W R R M 3 S O N 5 6 1 N 1 O

t0 03 (10) K L M N P t0 07 (20) < t0 03 (20) K z0 07 < t0 07 (20)6 Q @
: : : : x

t1
S 6

S 5

T U V

6

?

@ S &

t1
5
:

t0 03 (20) <
:

T

6

?

@

z1 6 ? @ z0 07

2

(n)

6

7

10 X i =1

X 2 =9 4 i P

Q

R

df

= 10

N

g

h

2. i

j

k

l n

m

E ) * 4 o

1. (a) X W

(2 2) (2 4) (2 6) (4 2) (4 4) (4 6) (6 2) (6 4) (6 6)
2 3 4 3 4 5 4 5 6

p

q

X4

o

S p 2

2 X = (X ; X )2 =(2 ; 1) i q

=1 S2 4

i

o

0 2 8

2 0 2

8 2 0

(b) X

x f (x)

2 1 9

3 2 9

4 3 9

5 2 9

6 1 9

(c) r

S2 g(S 2 )

0 3 9

2 4 9

8 2 9

4

7
3. (a) r s t u v w x

y

z

{

xw

x

y

z

{

S2 w

x

(2 4) (2 6) (4 2) ; (4 6) (6 2) (6 4) ; ; 3 4 ; 5 3 ; 4 5 ; 2 8 ; 2 2 ; 8 2
5 2 6

;

(b) r

x f (x)
(c) r

3 2 6

4 2 6

x

3 1 3

4 1 3

5 1 3

;!

f (x)

S2 g (S 2 )
5. (a) E (X ) =

2 4 6

8 2 6

S2

2 2 3

8 1 3

;!
2 9

g(S 2 )

X

xf (x) =

4 = 2 + 3 + 6 = 4 = E (X )

+ 6 + 12 + 10 + 6 = 4 9 9 9 9

V ar(X ) = E (X 2 ) ; E (X )2

= 4 + 18 + 48 + 50 + 36 ; 16 = 4 9 9 9 9 9 3 X (x ; 4)2 8 2 = =3 3 i 5

|

V ar(X ) =
4 9

2

n

3 =2=4 3

8

(b) E (S 2 ) = 2 7. (a) } u v ~ u

+8
~ w

2 9

=8= 3

2

w v

X
{ w

w

x

X X
(b) P

= = 1500 = =pn = 500 = 100
X > 1800

=P

5

Z>

1800 ; 1500 100

= P (Z > 3)
Z
1400 ; 1500 100

(c) P (1100

X

= 0:5 ; 0:4987 = 0:0013 1200 ; 1500 1400) = P 100 = P (;3
Z

;1) = 0:4987 ; 0:3413 = 0:1574

) df = 14 P (b < t < ;1:761) = 0:045 ) 0:05 ; P (t b) = 0:045 ) b = ;t0 005(14) = ;2:977 11. 2 = 32 (a) n = 15 df = 14 ! (n ; 1)S 2 > (n ; 1)75 = P 2 P (S > 75) = P
:

9. n = 15:

2

2

x2 (14) >

14 32

75

= 0:004
(b) P (S 2

< 15) = P

2

<

14 32

15

+ 1 ; 0:95 = 0:05

6

7
13. (a) w X

i

10

; 50

N (0 1 )

X=

16 X i X

i

(b)

w

= P (7:96 < X < 26:3) = P (X > 7:96) ; P (X > 26:3) = 0:95 ; 0:05 = 0: 9 16 X X ; X !2 2 X= (15) 10 i i

=1

10

; 50

2

2

(16)

=1

= P (7:26 < X < 25) = P (X > 7:26) ; P (X > 25) = 0:95 ; 0:05 = 0: 9
9 4

15.

X X ;Y
|

N 4 N

Y
25 4

N 7

16 4

;3

P (X < Y ) = P (X ; Y < 0) W

=X ;Y

N E X ;Y

V X ;Y

X ? Y : E ( W ) = E ( X ; Y ) = E (X ) ; E ( Y ) = 4 ; 7 = ;3 V (W ) = V (X ; Y ) = V (X ) + V (Y ) = P (X < Y ) = P (X ; Y < 0) = P (W < 0)
7
9 4

+ 16 = 25 4 4

0 B ( ) P B Wp; EWW ) @ V( )
) P (Z < 1:2) = 0:8849
17.

1 0 ; 3C r C A
25 4

(101 ; 1)S
2

}

s

2

X (101 ; 1) ! N (100 200) a P (S > 0:56) = P (S > 0:56
2

2

)=P

=P =P
19.

100S 2 0:52

> 125:44

!

100S 2 0:52

>

100

0:562 0:52

!

Z>

125:44 ; 100 p 200
2

= 100 50 0 1 B 430 ; 450 C P (X < 430) = P BZ < 100 C = P (Z < ;1:41) = 0:0793 @ A p 50 ! 2 N ;n 52 120 ; 49 a N = 12 n N ; 1 = 49 120 ; 1 21. X CLT 0 1 B 11 ; 12 < Z < r13 ; 12 C C P (11 < X < 13) = P B r @ A = 450 n a X CLT N

= P (Z > 1:80) = 0:0359 ! 2

= P (;1:81 < Z < 1:81) = 0:9298
8

25 49

71 119

25 49

71 119

7
23. X1

X2

)P
P p

p(1 ; p) N ; n a CLT N p = 0:02 n N ;1
10 300

X300

iid

Ber (p = 0:02)

0 10 BZ r 300 ; 0:02 =PB @ 0:02 0:98 700
300 999

= 0:023000:98 1000 ;;300 1000 1 1 C = P (Z 1:97) = 0:0244 C A

9