208 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
d. 500, 0.05, 0.95
25 5, 475 5
np q
np nq
== =
=≥ = ≥
28. 12, 0.67
8.04 5, 3.96 5
np
np nq
==
=≥ =<
Cannot use normal distribution because nq < 5.
a. (4) (0) (1) (2) (3)Px Px Px Px Px<= =+ =+ =+ =
()() ()() ()() ()()
012 111 210 39
12 0 12 1 12 2 12 3
0.67 0.33 0.67 0.33 0.67 0.33 0.67 0.33CCCC=+++
0.0036≈
29. 200, 0.34
68 5, 132 5
np
np nq
==
=≥ = ≥
Can use normal distribution.
a. 84.5 68 2.46
6.70
x
zµ
σ
−−
=≈ ≈
CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS 209
b. 65.5 68 0.37
6.70
x
zµ
σ
−−
=≈ ≈−
d. 6, 0.34
2.04 5, 3.96 5
np
np nq
==
=< =<
Cannot use normal distribution because np < 5 and nq < 5.
()()
60
66
( 6) 0.34 0.66 0.002Px C== ≈
0.999871
≈
b. (1)1(1)1(0)0.255Px Px Px≥=− <=− + ≈
c.
()() ()() ()()
010 19 28
10 0 10 1 10 2
(2)1(2)
1 0.029 0.971 0.029 0.971 0.029 0.971
0.003
Px Px
CCC
>=− ≤
≈− + +
≈
210 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
31. Binomial:
()() ()() ()()
511 610 79
16 5 16 6 16 7
(5 7) ( 5) ( 6) ( 7)
0.4 0.6 0.4 0.6 0.4 0.6
0.549
P x Px Px Px
CCC
≤≤==+=+=
=++
≈
32. Binomial:
()() ()() ()()
210 39 48
12 2 12 3 12 4
(2 4) ( 2) ( 3) ( 4)
0.5 0.5 0.5 0.5 0.5 0.5
0.191
P x Px Px Px
CCC
≤≤==+=+=
=++
≈
Normal: 6, 1.73np npqµσ== = ≈
33. 250, 0.70np==
60% say no 250(0.6) = 150 say no while 100 say yes.
100.5 175 10.28
7.25
x
zµ
σ
−−
=≈ ≈−
(less than or equal to 100 say yes) ( 100) ( 100.5) ( 10.28) 0PPxPxPz=≤ =< =<− ≈
It is highly unlikely that 60% responded no. Answers will vary.
34. 200, 0.11np==
9% of 200 = 18 people
18.5 22 0.79
4.42
x
zµ
σ
−−
=≈ ≈−
35. 100, 0.75np==
69.5 75 1.27
4.33
x
zµ
σ
−−
=≈ ≈−
CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS 211
CHAPTER 5 REVIEW EXERCISE SOLUTIONS
1. 15, 3µσ== 2. 3, 5µσ=− =
3. Curve B has the greatest mean because its line of symmetry occurs the farthest to the right.
4. Curve A has the greatest standard deviation because it is the most spread out.
6. 1.32 and 1.78 are unusual. 7. 0.6772
8. 0.2119 – 0.0094 = 0.2025 9. 0.6293
10. 0.0256 11. 1 – 0.2843 = 0.7157
12. 1 – 0.9994 = 0.0006 13. 0.00235
21. A: 8
B: 17
C: 23
D: 29
212 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
23. ( 1.28) 0.8997Pz<= 24. ( 0.74) 0.7704Pz>− =
25. ( 2.15 1.55) 0.9394 0.0158 0.9236Px−<< = − =
26. (0.42 3.15) 0.9992 0.6628 0.3364Pz<< = − =
27. ( 2.50 or 2.50) 2(0.0062) 0.0124Pz z<− > = =
28. ( 0 or 1.68) 0.5 0.0465 0.5465Pz z<>=+ =
31. 80 74 0.75
8
x
zµ
σ
−−
== =
( 80) ( 0.75) 1 ( 0.75) 1 0.7734 0.2266Px Pz Pz>= > =− < =− =
32. 71.6 74 0.3
8
x
zµ
σ
−−
== =−
( 71.6) ( 0.3) 1 ( 0.3) 1 0.3821 0.6179Px Pz Pz> = >− =− <− =− =
CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS 213
35. (a) 1900 2200 0.48
625
x
zµ
σ
−−
== =−
( 1900) ( 0.48) 0.3156Px Pz<=<−=
(b) 2000 2200 0.32
625
x
zµ
σ
−−
== =−
2500 2200 0.48
625
x
zµ
σ
−−
== =
(2000 2500) ( 0.32 0.48) 0.6844 0.3745 0.3099Px P z<< = − << = − =
(c) 2450 2200 0.4
625
x
zµ
σ
−−
== =
( 2450) ( 0.4) 0.3446Px Pz>=>=
37. No, none of the events are unusual because their probabilities are greater than 0.05.
38. Yes, the event in part (c) is unusual because its probability is less than 0.05.
39. 0.07z=− 40. 1.28z=− 41. 1.13z=
42. 2.05z=− 43. 1.04z= 44. 0.10z=−
214 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
49. 95th percentile ⇒ Area = 0.95 ⇒ 1.645z=
()()
48 1.645 2.2 51.6xzµσ=+ = + ≈ meters
51. Top 10% ⇒ Area = 0.90 ⇒ 1.28z=
()()
48 1.28 2.2 50.8xzµσ=+ = + ≈ meters
53. {90 90 90, 90 90 120, 90 90 160, 90 90 210, 90 120 90, 90 120 120, 90 120 160, 90 120 210,
90 160 90, 90 160 120, 90 160 160, 90 160 210, 90 210 90, 90 210 120, 90 210 160, 90 210 210,
120 90 90, 120 90 120, 120 90 160, 120 90 210, 120 120 90, 120 120 120, 120 120 160,
120 120 210, 120 160 90, 120 160 120, 120 160 160, 120 160 210, 120 210 90, 120 210 120,
54. {00, 01, 02, 03, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33}
1.5, 1.118µσ=≈
1.118
1.5, 0.791
2
xx
µσ=≈≈
The means are the same, but x
σ is less than σ.
CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS 215
56. 35.1
108.3, 5.550
40
xx
n
σ
µσ===≈
(2000 2500) ( 1.11 1.66) 0.9515 0.1335 0.8180Px P z<< = − << = − =
(c) 2450 2200 250 1.39
625 180.42
12
x
z
n
µ
σ
−−
== ≈ ≈
58. (a) 1.0 1.5 0.5 5.29
0.25 0.0945
7
x
z
n
µ
σ
−−−
== ≈ ≈−
2.0 1.5 0.5 5.29
0.25 0.0945
7
x
z
n
µ
σ
−−
== ≈ ≈
216 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
(c) 2.2 1.5 0.7 7.41
0.25 0.0945
7
x
z
n
µ
σ
−−
== ≈ ≈
(2.2) (7.41)0Px Pz>=> ≈
(a) is larger and (b) and (c) are smaller.
60. (a) 1400 1300 100 2.4
250 41.67
36
x
z
n
µ
σ
−−
== ≈ ≈
( 1400) ( 2.4) 0.9918Px Pz<=<=
(b) 1150 1300 150 3.6
250 41.67
36
x
z
n
µ
σ
−−−
== ≈ ≈−
( 1150) ( 3.6) 1 0 1Px Pz>=>−≈−=
62. Assuming the distribution is normally distributed:
525 500 25 3.23
30 7.75
15
x
z
n
µ
σ
−−
== ≈≈
( 525) ( 3.23) 1 ( 3.23) 1 0.9994 0.0006Px Pz Pz>=> =−< =− =
CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS 217
65. (25) (24.5)Px Px≥= > 65. ( 36) ( 36.5)Px Px≤= <
67. ( 45) (44.5 45.5)Px P x== << 68. ( 50) (49.5 50.5)Px P x== <<
70. 15, 0.31np==
4.65 5np =<, 10.35 5nq =≥
Cannot use normal distribution because np < 5.
(8)1(8)Px Px>=− ≤
[
]
()() ()() ()()
015 114 87
15 0 15 1 15 8
1(0)(1) (8)
1 0.31 0.69 0.31 0.69 0.31 0.69
0.019
Px Px Px
CC C
=− = + = + + =
⎡⎤
=− + + +
⎢⎥
⎣⎦
≈
…
…
CHAPTER 5 QUIZ SOLUTIONS
218 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
2. (a) 5.36 5.5 1.75
0.08
x
zµ
σ
−−
== =−
5.64 5.5 1.75
0.08
x
zµ
σ
−−
== =
3. 125 100 1.67
15
x
zµ
σ
−−
== ≈
( 125) ( 1.67) 0.0475Px Pz>=> =
Yes, the event is unusual because its probability is less than 0.05.
5. ( 112) ( 0.80) 0.2119 21.19%Px Pz>=> = →
6. 90 100 0.67
15
x
zµ
σ
−−
== ≈−
( 90) ( 0.67) 0.2514Px Pz<= <− =
(2000)(0.2514)=502.8 503 students
CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS 219
9. 105 100 5 2.58
15 1.936
60
x
z
n
µ
σ
−−
== ≈≈
10. 105 100 0.33
15
x
zµ
σ
−−
== ≈
( 105) ( 0.33) 0.3707Px Pz>=> =
105 100 5 1.29
15 3.873
15
x
z
n
µ
σ
−−
== ≈≈
( 105) ( 1.29) 0.0985Px Pz>=> ≈
You are more likely to select one student with a test score greater than 105 because the standard
error of the mean is less than the standard deviation.
CUMULATIVE REVIEW, CHAPTERS 3-5
1. (a)
()
()
50 0.15 7.5 5
50 0.85 42.5 5
np
nq
==≥
==≥
220 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
2.
x P(x) xP(x)
x
µ−
(
)
2
xµ−
(
)
2
xµ−P(x)
2 0.427 0.854 –1.13 1.277 0.545
3 0.227 0.681 –0.13 0.017 0.004
4 0.200 0.800 0.87 0.757 0.151
5 0.093 0.465 1.87 3.497 0.325
6 0.034 0.204 2.87 8.237 0.280
7 0.018 0.126 3.87 14.977 0.270
() 3.13xP x =
∑
()
2( ) 1.575xPxµ−≈
∑
3.
x P(x) xP(x)
x
µ−
(
)
2
xµ−
(
)
2
xµ−P(x)
0 0.012 0.000 –3.596 12.931 0.155
1 0.049 0.049 –2.596 6.739 0.330
2 0.159 0.318 –1.596 2.547 0.405
3 0.256 0.768 –0.596 0.355 0.091
4 0.244 0.976 0.404 0.163 0.040
5 0.195 0.975 1.404 1.971 0.384
6 0.085 0.510 2.404 5.779 0.491
() 3.596xP x =
∑
()
2( ) 1.896xPxµ−≈
∑
(a) () 3.6xP xµ=≈
∑
CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS 221
6. 0.7642 7. 0.0010
8. 1 0.2005 0.7995−= 9. 0.9984 0.500 0.4984−=
10. 0.3974 0.1112 0.2862−= 11.
()
0.5478 1 0.9573 0.5905+− =
12. 11, 0.45np==
(a) P(8) = 0.0462; unusual because the probability is less than 0.05.
(b) ( 5) 0.6029Px≥=
(c) ( 2) 0.0139Px<= ; unusual because the probability is less than 0.05.
14. (a) 0.2777
(b) 0.8657
(c) Dependent. P(being a public school teacher having 20 years or more of full-time teaching
experience) P(being a public school teacher)
(d) 0.8799 0.0612 0.0530 0.8881+−=
(e) 0.3438 0.1201 0.0462 0.4177+−=
222 CHAPTER 5 │ NORMAL PROBABILITY DISTRIBUTIONS
16. (a) 36 44
( 36) ( 1.6) 0.0548
5
Px Pz Pz
⎛⎞
−⎟
⎜
<= < = <− =
⎟
⎜⎟
⎜
⎝⎠
(b) 42 44 60 44
(42 60) 55
Px P z
⎛⎞
−−
⎟
⎜
<< = << ⎟
⎜⎟
⎜
⎝⎠
(0.4 3.2)
0.9993 0.3446 0.6547
Pz=− <<
=−=
(c) Top 5% 1.645z⇒=
44 (1.645)(5) 52.2xzµσ=+ = + ≈ months