Chapter 8
8.1a. P(X > 45)
1550
2)6075(
1550
2)4560(
−
+
−
= .0800
b. P(10 < X < 40)
1550
2)6065(
2)4560(
+
6)3040(
7)1530(
17)1015(
−
+
−
+
−
= .3333
8.2a. P(X > 45)
1550
3)6075(
1550
3)4560(
−
+
−
= .1200
b. P(10 < X < 40)
1550
8)3040(
1550
8)1530(
1550
16)1015(
−
+
−
+
−
= .4800
8.3a. P(55 < X < 80)
= .6167
b. P(X > 65)
1060
1)90100(
1060
7)8090(
1060
24)7080(
1060
5)6570(
−
+
−
+
−
+
−
= .5750
8.4 a.
b. P(X > 25) = 0
c.
8.8
3)3060(10.=−
; The top decile = 60–3 = 57 minutes
8.9 f(x) =
65
1
)110175(
1=
−
110 < x < 175
For Exercises 8.11 to 8.14 we calculate probabilities by determining the area in a triangle. That
is,
Area in a triangle = (.5)(height)(base)
8.11a.
b. P(0 < X < 2) = (.5)(2–0)(1) = 1.0
8.12 a
8.13a.
b. P(1 < X < 3) = P(X < 3) – P(X < 1) =
)01(
25
1
2
1
)03(
25
3
2
1−−−
= .18 – .02 = .16
c. P(4 < X < 8) = P(4 < X < 5) + P(5 < X < 8)
P(4 < X < 5)= P(X < 5) – P(X <4) =
)04(
25
4
2
1
)05(
25
5
2
1−−−
= .5 – .32 = .18
8.14 a. f(x) = .10 – .005x 0
x
20
b. P(X > 10) = (.5)(.05)(20–10) = .25
c. P(6 < X < 12) = P(X > 6) – PX > 12) = (.5)(.07)(20–6) – (.5)(.04)(20–12) = .49 – .16 = .33
8.15 P( Z < 1.50) = .9332
8.19 P(Z < −1.60) = .0548
8.20 P(Z < − 2.30) = .0107
8.24 P(Z > 1.67) = 1 – P(Z < 1.67) = 1 – .9525 = .0475
8.25 P(Z < 2.84) = ..9977
8.26 P(1.14 < Z < 2.43) = P(Z < 2.43) – P(Z < 1.14) = .9925 – .8729 = .1196
8.27 P(–0.91 < Z < –0.33) = P(Z < −.33) – P(Z < −.91) = .3707 – .1814 = .1893
8.31 P(Z <
02.
z
) = 1 – .02 = .9800;
02.
z
= 2.05
8.32 P(Z <
045.
z
) = 1 – .045 = .9550;
045.
z
20.
z
20.
z
15.
z
= 1.70
8.36 P(800 < X < 1100) =
−
−
−
250
000,1100,1
X
250
000,1800
P
−
2.2
P
= P(–.8 < Z < .4)
= P(Z < .4) − P(Z < −.8) = .6554 − .2119 = .4435
c P(X < 4) =
−
−
2.2
3.64X
P
= P(Z < –1.05) = .1469
8.39 P(Z <
10.
z
) = 1 – .10 = .9000;
10.
z
= 1.28;
−
=x
z10.
;
2.2
3.6x
28.1 −
=
; x = 9.116
Calls last at least 9.116 minutes.
8.40 P(X > 5,000) =
−
−
200
100,5000,5
X
P
= P(Z > –.5) = 1 −P(Z < −.5) = 1 − .3085 = .6915
8.43 P(Z <
001.
z
) = .9990;
001.
z
= 3.08;
−
=x
z001.
;
400,2
000,10x
08.3 −
=
; x = 17,392
b P(X < 60) =
−
−
4
6560X
P
= P(Z < –1.25) = .1056
8.46 Top 5%: P(Z <
05.
z
) = 1 – .05 = .9500;
05.
z
= 1.645;
−
=x
z05.
;
5.1
32x
645.1 −
=
; x =
34.4675
Bottom 5%: P(Z <
05.
z−
) = .0500;
;645.1z 05.−=−
−
=− x
z05.
;
5.1
32x
645.1 −
=−
;
x = 29.5325
8.48 P(X > 8) =
−
−
667.
2.78X
P
25.
z
25.
z
= P (Z > 1.20) = 1 – P(Z < 1.20) = 1 – .8849 = .1151
b P(7 < X < 9) =
−
−
−
1.2
5.79
X
1.2
5.77
P
z05.
= P(–.24 < Z < .71)
= P(Z < .71) − P(0 < Z < −.24) = .7611 − .4052 = .3559
.2643
b P(X < 10,000) =
−
−
800
500,11000,10
X
P
= P(Z < –1.88) = .0301
8.52 P(Z < –
01.
z
) = .0100;
05.
z−
= –2.33;
−
=− x
z01.
;
800
500,11x
33.2 −
=−
−
2624X
P
; x = 9,636
8.54 a P(X > 30) =
−
−
7
2730X
P
= P(Z > .43) = 1 – P(Z < .43) = 1 – .6664 = .3336
b P(X > 40) =
−
−
7
2740
X
P
= P(Z > 1.86) = 1 – P(Z < 1.86) = 1 – .9686 = .0314
c P(X < 15) =
−
−
7
2715X
P
= P(Z < –1.71) = .0436
−
27x
2.1
b P(7 < X < 10) =
−
−
−
2.1
5.710
X
2.1
5.77
P
10.
z
10.
z
−
=− x
z10.
= P(–.42 < Z < 2.08)
8.57 A: P(Z <
10.
z
) = 1– .10 = .9000;
10.
z
= 1.28;
−
=x
z10.
;
10
70x
28.1 −
=
; x = 82.8
B: P(Z <
40.
z
) = 1 –.40 = .6000;
40.
z
= .25;
−
=x
z40.
;
10
70x
25.−
=
; x = 72.5
C: P(Z < –
20.
z
) = .2000;
;84.z 20.−=−
−
=− x
z20.
;
10
70x
84.−
=−
05.
z
−
z05.
02.
z
02.
z
; x = 61.6;
8.60 P(X < 45,000) =
 −
−
444,13
825,41000,45
X
P
= P(Z < .24) = .5948
8.61 P(Z <
01.
z−
) = .0100;
;33.2z 01.−=−
−
=− x
z01.
;
8
75x
33.2 −
=−
; x = 56.36
8.63 P(Z <
06.
z
) = 1 – .06 = .9400;
06.
z
= 1.55;
−
=ROP
z06.
;
30
200ROP
55.1 −
=
20.
z
z20.
30.
z
30.
z
−
=x
z30.
90
52.−
;
ROP = 246.5 (rounded to 247)
8.66 P( Z <
40.
z
) = 1– .40 = .6000;
40.
z
= .25;
−
=x
z40.
;
90
850x
25.−
=
; .x = 872.5
(rounded to 873)
8.67 From Exercise 7.57:
= 65,
2
= 21, and
= 4.58
8.69 a. P(X > 25) =
−
−
18
1425
X
P
= P(Z > .61) = 1 – P(Z < .61) = 1 – .7291 = .2709
b. P(X < 0) =
−
−
18
140
X
P
= P(Z < –.78) = .2177
8.71
8.72
8.73 a
5..)1(5.. ee)1X(P −− ==
= .6065
a
2..)4(.5.. ee)4.X(P −− ==
= .8187
2
2.5
3
3.5
Exponential Distribution (Lambda = 3)
0.2
0.25
0.3
Exponential Distribution (Lambda = .25)
c
6.3..)2(3..)1(3.. eeee)2X1(P −−−− −=−=
= .7408 – .5488 = .1920
d P(X = 3) = 0
8.77
= 10 trucks/hour = .167 truck/minute
5.2)15(167. ee)15X(P −− ==
=/1
=/1
= .0821
8.80
=/1
= 7.5 minutes;
= .133 service/minute
665.)5(133.e1e1)5X(P −− −=−=
= 1– .5143 = .4857
8.83 a 1.341 b 1.319 c 1.988 d 1.653
8.84 a 2.750 b 1.282 c 2.132 d 2.528
8.87 a .0189 b .0341 c .0927 d .0324
8.88 a .1744 b .0231 c .0251 d .0267
8.93 a .2688 b 1.0 c .9903 d 1.0
8.94 a .4881 b .9158 c .9988 d .9077
8.99 a .0510 b .1634 c .0222 d .2133
8.100 a .1050 b .1576 c .0001 d .0044