Eco249Answers8-11.doc

Chapter 8
8.1a. P(X > 45) [pic]= .0800
b. P(10 < X < 40) [pic]= .4800
c. P(X < 25) [pic]= .7533
d. P(35 < X < 65) [pic]= .1333 8.2a. P(X > 45) [pic]= .1200
b. P(10 < X < 40) [pic]= .3333
c. P(X < 25) [pic]= .6667
d. P(35 < X < 65) [pic]= .1867 8.3a. P(55 < X < 80) [pic]= .6167
b. P(X > 65) [pic]= .5750
c. P(X < 85) [pic]= .9250
d. P(75 < X < 85) [pic]= .2583 8.4 a.
[pic]
b. P(X > 25) = 0
c. P(10 < X < 15) = [pic] = .25
d. P(5.0 < X < 5.1) = [pic] = .005 8.5a. f(x) = [pic]= [pic] 20 < x < 60
[pic]
b. P(35 < X < 45) = (45-35) [pic] = .25
c.
[pic]
8.6 f(x) = [pic] 30 < x < 60
a. P(X > 55) = [pic]= .1667
b. P(30 < X < 40) = [pic]= .3333
c. P(X = 37.23) = 0 8.7 [pic]; The first quartile = 30 + 7.5 = 37.5 minutes 8.8 [pic]; The top decile = 60-3 = 57 minutes 8.9 f(x) = [pic] 110 < x < 175
a. P(X > 150) = [pic]= .3846
b. P(120 < X < 160) = [pic]= .6154 8.10 .20(175-110) = 13. Bottom 20% lie below (110 + 13) = 123
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.
[pic]
b. P(0 < X < 2) = (.5)(2-0)(1) = 1.0
c. P(X > 1) = (.5)(2 - 1)(.5) = .25
d. P(X < .5) = 1 - P(X > .5) = 1 - (.5)(.75)(2-.5) = 1 - .5625 = .4375
e. P(X = 1.5) = 0 8.12 a
[pic]
b. P(2 < X < 4) = P(X < 4) - P(X < 2) = (.5)(3/8)(4-1) - (.5)(1/8)(2-1) =
.5625 - .0625 = .5
c. P(X < 3) = (.5)(2/8)(3-1) = .25 8.13a.
[pic]
b. P(1 < X < 3) = P(X < 3) - P(X < 1) = [pic]= .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 5) - P(X > 8) = [pic]= .5 - .08 = .42
P(4 < X < 8) = .18 + .42 = .60
d. P(X < 7) = 1 - P(X > 7)
P(X > 7) = [pic]= .18
P(X < 7) = 1 - .18 = .82
e. P(X > 3) = 1 - P(X < 3)
P(X < 3) = [pic]= .18
P(X > 3) = 1 - .18 = .82 8.14 a. f(x) = .10 - .005x 0 [pic] x [pic] 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 8.16 P(Z < 1.51) = .9345 8.17 P(Z < 1.55) = .9394 8.18 P(Z < -1.59) = .0559
8.19 P(Z < -1.60) = .0548 8.20 P(Z < - 2.30) = .0107 8.21 P(-1.40 < Z < 0.60) = P( Z < 0.60) - P(Z < -1.40) = .7257- .0808 =
.6449 8.22 P(Z > -1.44) = 1 - P(Z < -1.44) = 1 - .0749 = .9251 8.23 P(Z < 2.03) = .9788 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 8.28 P(Z > 3.09) = .5 - P(0 < Z < 3.09) = .5 - .4990 = .0010 8.29 P(Z > 0) = 1 - P(Z < 0) = 1 - .5 = .5 8.30 P(Z > 4.0) = 0 8.31 P(Z < [pic]) = 1 - .02 = .9800; [pic]= 2.05 8.32 P(Z 145) = [pic]= P(Z > 2.25) = 1 - P(Z < 2.25) = 1 - .9878 = .0122 8.35 P(Z < [pic]) = 1 - .15 = .8500; [pic]= 1.04; [pic]; [pic]; x = 291.6 8.36 P(800 < X < 1100) = [pic]= P(-.8 < Z < .4)
= P(Z < .4) - P(Z < -.8) = .6554 - .2119 = .4435 8.37 P(Z < [pic]) = .0800; [pic][pic]; [pic]; x = 38.72 8.38 a P(5 < X < 10) = [pic]= P(-.59 < Z > 1.68)
= P(Z < 1.68) - P(Z < -.59) = .9535 - .2776 = .6759
b P(X > 7) = [pic]= P(Z > .32) = 1 - P(Z < .32) = 1 - .6255 = .3745
c P(X < 4) = [pic]= P(Z < -1.05) = .1469 8.39 P(Z < [pic]) = 1 - .10 = .9000; [pic]= 1.28; [pic]; [pic]; x = 9.116
Calls last at least 9.116 minutes. 8.40 P(X > 5,000) = [pic]= P(Z > -.5) = 1 -P(Z < -.5) = 1 - .3085 = .6915 8.41 P(Z 12,000) = [pic]= P(Z > .83) = 1 - P(Z < .83) = 1 - .7967 =
.2033
b P(X < 9,000) = [pic]= P(Z < -.42) = .3372 8.43 P(Z < [pic]) = .9990; [pic]= 3.08; [pic]; [pic]; x = 17,392 8.44 a P(X > 70) = [pic]= P(Z > 1.25) = 1 - P(Z < 1.25) = 1 - .8944 = .1056
b P(X < 60) = [pic]= P(Z < -1.25) = .1056
c P(55 < X < 70) = [pic]= P(-2.50 < Z < 1.25)
= P(Z < 1.25) - P(Z < -2.50) = .8944- .0062 = .8882 8.45 a P(X < 70,000) = [pic]= P(Z < -1.88) = .0301
b P(X > 100,000) = [pic]= P(Z > 2.81) = 1 - P(Z < 2.81) = 1 - .9975 =
.0025 8.46 Top 5%: P(Z < [pic]) = 1 - .05 = .9500; [pic]= 1.645; [pic]; [pic]; x
= 34.4675
Bottom 5%: P(Z < [pic]) = .0500; [pic][pic]; [pic];
x = 29.5325 8.47 a P(X > 36) = [pic]= P(Z > 2.67) = 1 - P(Z < 2.67) = 1 - .9962 = .0038
b P(X < 34) = [pic]= P(Z < 1.33) = .9082
c P(30 < X < 33) = [pic]= P(-1.33 < Z < .67)
= P(Z < .67) - P(Z < -1.33) = .7486 - .0918 = .6568 8.48 P(X > 8) = [pic]= P (Z > 1.20) = 1 - P(Z < 1.20) = 1 - .8849 = .1151 8.49 P(Z < [pic]) = .7500; [pic] = .67; [pic]; [pic]; x = 7.65 hours 8.50 a P(X > 10) = [pic]= P Z > 1.19) = 1 - P(Z < 1.19) = 1 - .8830 = .1170 b P(7 < X < 9) = [pic]= P(-.24 < Z < .71)
= P(Z < .71) - P(0 < Z < -.24) = .7611 - .4052 = .3559
c P(X < 3) = [pic]= P Z < -2.14) = .0162
d P(Z < -[pic]) = .0500; [pic] = -1.645; [pic]; [pic]; x = 4.05 hours 8.51 a P(X > 12,000) = [pic]= P Z > .63) = 1- P(Z < .63) = 1 - .7357 =
.2643
b P(X < 10,000) = [pic] = P(Z < -1.88) = .0301 8.52 P(Z < -[pic]) = .0100; [pic] = -2.33;[pic]; [pic]; x = 9,636 8.53 a P(24 < X < 28) = [pic] = P(-.80 < Z < .80) = P(Z < .80) - P(Z <
-.80)
= .7881 - .2119 = .5762
b P(X > 28) =[pic] = P(Z > .80) = 1 - P(Z < .80) = 1 - .7881 = .2119
c P(X < 24) =[pic] = P(Z < -.80) =.2119 8.54 a P(X > 30) = [pic]= P(Z > .43) = 1 - P(Z < .43) = 1 - .6664 = .3336
b P(X > 40) = [pic]= P(Z > 1.86) = 1 - P(Z < 1.86) = 1 - .9686 = .0314
c P(X < 15) = [pic]= P(Z < -1.71) = .0436
d P(Z < [pic]) = 1 - .20 = .8000; [pic] = .84; [pic]; [pic]; x = 32.88 8.55 a P(X < 4) = [pic]= P(Z < -2.92) = .0018
b P(7 < X < 10) = [pic]= P(-.42 < Z < 2.08) = P(Z < 2.08) - P(Z < -.42)
= .9812 - .3372 = .6440 8.56 a P(X < 10) = [pic] = P(Z < -2.33) = .0099
b P(Z < -[pic]) = .1000; -[pic] = -1.28; [pic]; [pic]; x = 12.88 8.57 A: P(Z < [pic]) = 1- .10 = .9000; [pic] = 1.28; [pic]; [pic]; x =
82.8
B: P(Z < [pic]) = 1 -.40 = .6000; [pic] = .25; [pic]; [pic]; x = 72.5
C: P(Z < -[pic]) = .2000; [pic][pic]; [pic]; x = 61.6;
D: P(Z < -[pic]) = .0500; [pic][pic]; [pic]; x = 53.55 8.58 P(Z < [pic]) = 1 - .02 = .9800; [pic] = 2.05; [pic]; [pic]; x =
132.80 (rounded to 133) 8.59 P(X > 70,000) = [pic] = P(Z > .47) = 1 - P(Z < .47) = 1 - .6808 =
.3192 8.60 P(X < 45,000) = [pic] = P(Z < .24) = .5948 8.61 P(Z < [pic]) = .0100; [pic] [pic]; [pic]; x = 56.36 8.62 P(x > 150,000) = [pic]= P(Z > 1.68) = 1 - P(Z < 1.68) = 1 - .9535 =
.0465 8.63 P(Z < [pic]) = 1 - .06 = .9400; [pic] = 1.55; [pic]; [pic]; ROP =
246.5 (rounded to 247) 8.64 P(Z < [pic]) = 1 - .20 = .8000; [pic] = .84; [pic]; [pic]; x = 171 8.65 P(Z < [pic]) = 1 - .30 = .7000; [pic] = .52; [pic]; [pic]; x = 896.8 (rounded to 897) 8.66 P( Z < [pic]) = 1- .40 = .6000; [pic] = .25; [pic]; [pic]; .x =
872.5
(rounded to 873) 8.67 From Exercise 7.57: [pic] = 65, [pic] = 21, and [pic] = 4.58
P(X > 60) = [pic]= P(Z > -1.09) = 1 - P(Z < -1.09) = 1 -.1379 = .8621 8.68 P(X < 150) = [pic]= P(Z < .90) = .8159 8.69 a. P(X > 25) = [pic]= P(Z > .61) = 1 - P(Z < .61) = 1 - .7291 = .2709
b. P(X < 0) = [pic]= P(Z < -.78) = .2177 8.70 a. P(X < 0) = [pic]= P(Z < -.73) = .2327
b. P(X > 20) = [pic]= P(Z > .65) = 1 - P(Z < .65) = 1 - .7422 = .2578
8.71
[pic] 8.72
[pic] 8.73 a [pic]= .6065
a [pic]= .8187
c [pic]= 1 - .7788 = .2212
d [pic]= 1 - .3679 = .6321 8.74 a [pic]= .5488
b [pic]= 1 - .3012 = .6988
c [pic]= .7408 - .5488 = .1920
d P(X = 3) = 0 8.75 [pic]= 6 kilograms/hour = .1 kilogram/minute
[pic]= .2231 8.76 [pic]hours; [pic]= .04 breakdowns/hour
[pic]= .1353 8.77 [pic]= 10 trucks/hour = .167 truck/minute
[pic]= .0821 8.78 [pic]= 5 minutes; [pic]= .2 customer/minute
[pic]= 1- .1353 = .8647 8.79 [pic]= 2.7 minutes; [pic]= .37 service/minute
[pic]= 1- .3296 = .6704 8.80 [pic]= 7.5 minutes; [pic]= .133 service/minute
[pic]= 1- .5143 = .4857 8.81 [pic]= 125 seconds; [pic]= .008 transactions/second = .48
transactions/minute
[pic]= .2369 8.82 [pic]= 6 minutes; [pic]= .167 customers/minute
[pic]= .1889 8.83 a 1.341 b 1.319 c 1.988 d 1.653 8.84 a 2.724 b 1.282 c 2.132 d 2.528 8.85 a 1.3406 b 1.3195 c 1.9890 d 1.6527 8.86 a 1.6556 b 2.6810 c 1.9600 d 1.6602
8.87 a .0189 b .0341 c .0927 d .0324 8.88 a .1744 b .0231 c .0251 d .0267 8.89 a 9.24 b 136 c 9.39 d 37.5 8.90 a 17.3 b 50.9 c 2.71 d 53.5 8.91 a 73.3441 b 102.946 c 16.3382 d 24.7690 8.92 a 33.5705 b 866.911 c 24.3976 d 261.058 8.93 a .2688 b 1.0 c .9903 d 1.0 8.94 a .4881 b .9158 c .9988 d .9077 8.95 a 4.35 b 8.89 c 3.29 d 2.50 8.96 a 2.84 b 1.93 c 3.60 d 3.37 8.97 a 1.4857 b 1.7633 c 1.8200 d 1.1587 8.98 a 1.5204 b 1.5943 c 2.8397 d 1.1670 8.99 a .0510 b .1634 c .0222 d .2133 8.100 a .1050 b .1576 c .0001 d .0044 Chapter 9
9.1a. 1/6
b. 1/6 9.2 a [pic]=P(1,1)= 1/36
b [pic]= P(6,6) = 1/36 9.3a P([pic] = 1) = (1/6)[pic]= .0001286
b P([pic] = 6) = (1/6)[pic]= .0001286 9.4 The variance of [pic]is smaller than the variance of X. 9.5 The sampling distribution of the mean is normal with a mean of 40 and a
standard deviation of 12/[pic]= 1.2. 9.6 No, because the sample mean is approximately normally distributed. 9.7 a [pic]= [pic] = P(Z > 1.00) = 1 - P(Z < 1.00) = 1 - .8413 = .1587
b [pic][pic] = P(Z < -.80) = .2119
c [pic][pic] = P(Z > 2.00) = 1 - P(Z < 2.00) = 1 - .9772 = .0