05 ch ken black solution

Chapter 5: Discrete Distributions 1

Chapter 5 Discrete Distributions

LEARNING OBJECTIVES

The overall learning objective of Chapter 5 is to help you understand a category of probability distributions that produces only discrete outcomes, thereby enabling you to:

1. Distinguish between discrete random variables and continuous random variables. 2. Know how to determine the mean and variance of a discrete distribution. 3. Identify the type of statistical experiments that can be described by the binomial

distribution and know how to work such problems. 4. Decide when to use the Poisson distribution in analyzing statistical experiments and

know how to work such problems. . 5. Decide when binomial distribution problems can be approximated by the Poisson

distribution and know how to work such problems. 6. Decide when to use the hypergeometric distribution and know how to work such

problems

CHAPTER TEACHING STRATEGY

Chapters 5 and 6 introduce the student to several statistical distributions. It is important

to differentiate between the discrete distributions of chapter 5 and the continuous distributions of chapter 6.

The approach taken in presenting the binomial distribution is to build on techniques

presented in chapter 4. It can be helpful to take the time to apply the law of multiplication for independent events to a problem and demonstrate to students that sequence is important. From there, the student will more easily understand that by using combinations, one can more quickly determine the number of sequences and weigh the probability of obtaining a single sequence by that number. In a sense, we are developing the binomial formula through an inductive process.


Thus, the binomial formula becomes more of a summary device than a statistical "trick". The binomial tables presented in this text are noncumulative. This makes it easier for the student to recognize that the table is but a listing of a series of binomial formula computations. In addition, it lends itself more readily to the graphing of a binomial distribution.

It is important to differentiate applications of the Poisson distribution from binomial

distribution problems. It is often difficult for students to determine which type of distribution to apply to a problem. The Poisson distribution applies to rare occurrences over some interval. The parameters involved in the binomial distribution (n and p) are different from the parameter (Lambda) of a Poisson distribution.

It is sometimes difficult for students to know how to handle Poisson problems where the

interval for the problem is different than the interval for which Lambda was developed. If they can view Lambda as a long run average which can be appropriately adjusted for various intervals, then students can be more successful with these types of problems.

Solving for the mean and standard deviation of binomial distributions prepares the

students for chapter 6 where the normal distribution is sometimes used to solve binomial distribution problems. In addition, graphing binomial and Poisson distributions affords the student the opportunity to visualize the meaning and impact of a particular set of parameters for a distribution.

It can be emphasized that the hypergeometric distribution is an exact distribution.

However, it is cumbersome to determine probabilities using the hypergeometric formula particularly when computing cumulative probabilities. The hypergeometric distribution can be presented as a fall-back position to be used when the binomial distribution should not be applied because of the non independence of trials and size of sample.

CHAPTER OUTLINE

5.1 Discrete Versus Continuous Distributions 5.2 Describing a Discrete Distribution Mean, Variance, and Standard Deviation of Discrete Distributions Mean or Expected Value Variance and Standard Deviation of a Discrete Distribution

5.3 Binomial Distribution Solving a Binomial Problem Using the Binomial Table Using the Computer to Produce a Binomial Distribution Mean and Standard Deviation of the Binomial Distribution Graphing Binomial Distributions


5.4 Poisson Distribution Working Poisson Problems by Formula Using the Poisson Tables Mean and Standard Deviation of a Poisson Distribution Graphing Poisson Distributions Using the Computer to Generate Poisson Distributions Approximating Binomial Problems by the Poisson Distribution

5.5 Hypergeometric Distribution

Using the Computer to Solve for Hypergeometric Distribution Probabilities

KEY TERMS

Binomial Distribution Hypergeometric Distribution Continuous Distributions Lambda (λ) Continuous Random Variables Mean, or Expected Value Discrete Distributions Poisson Distribution Discrete Random Variables Random Variable

SOLUTIONS TO PROBLEMS IN CHAPTER 5

5.1 x P(x) x·P(x) (x-µ)2 (x-µ)2·P(x) 1 .238 .238 2.775556 0.6605823 2 .290 .580 0.443556 0.1286312 3 .177 .531 0.111556 0.0197454 4 .158 .632 1.779556 0.2811700 5 .137 .685 5.447556 0.7463152 µ = [x·P(x)] = 2.666 σ2 = (x-µ)2·P(x) = 1.836444

σ = 836444.1 = 1.355155


5.2 x P(x) x·P(x) (x-µ)2 (x-µ)2·P(x) 0 .103 .000 7.573504 0.780071 1 .118 .118 3.069504 0.362201 2 .246 .492 0.565504 0.139114 3 .229 .687 0.061504 0.014084 4 .138 .552 1.557504 0.214936 5 .094 .470 5.053504 0.475029 6 .071 .426 10.549500 0.749015 7 .001 .007 18.045500 0.018046 µ = [x·P(x)] = 2.752 σ2 = (x-µ)2·P(x) = 2.752496

σ = 752496.2 = 1.6591 5.3 x P(x) x·P(x) (x-µ)2 (x-µ)2·P(x) 0 .461 .000 0.913936 0.421324 1 .285 .285 0.001936 0.000552 2 .129 .258 1.089936 0.140602 3 .087 .261 4.177936 0.363480 4 .038 .152 9.265936 0.352106 E(x)=µ= [x·P(x)]= 0.956 σ2 = (x-µ)2·P(x) = 1.278064

σ = 278064.1 = 1.1305

5.4 x P(x) x·P(x) (x-µ)2 (x-µ)2·P(x) 0 .262 .000 1.4424 0.37791 1 .393 .393 0.0404 0.01588 2 .246 .492 0.6384 0.15705 3 .082 .246 3.2364 0.26538 4 .015 .060 7.8344 0.11752 5 .002 .010 14.4324 0.02886 6 .000 .000 23.0304 0.00000 µ = [x·P(x)] = 1.201 σ2 = (x-µ)2·P(x) = 0.96260

σ = 96260. = .98112


5.5 a) n = 4 p = .10 q = .90

P(x=3) = 4C3(.10)3(.90)1 = 4(.001)(.90) = .0036

b) n = 7 p = .80 q = .20

P(x=4) = 7C4(.80)4(.20)3 = 35(.4096)(.008) = .1147

c) n = 10 p = .60 q = .40

P(x > 7) = P(x=7) + P(x=8) + P(x=9) + P(x=10) =

10C7(.60)7(.40)3 + 10C8(.60)8(.40)2 + 10C9(.60)9(.40)1 +10C10(.60)10(.40)0 =

120(.0280)(.064) + 45(.0168)(.16) + 10(.0101)(.40) + 1(.0060)(1) = .2150 + .1209 + .0403 + .0060 = .3822

d) n = 12 p = .45 q = .55 P(5 < x < 7) = P(x=5) + P(x=6) + P(x=7) = 12C5(.45)5(.55)7 + 12C6(.45)6(.55)6 + 12C7(.45)7(.55)5 = 792(.0185)(.0152) + 924(.0083)(.0277) + 792(.0037)(.0503) = .2225 + .2124 + .1489 = .5838

5.6 By Table A.2:

a) n = 20 p = .50

P(x=12) = .120

b) n = 20 p = .30

P(x > 8) = P(x=9) + P(x=10) + P(x=11) + ...+ P(x=20) =


.065 + .031 + .012 + .004 + .001 + .000 = .113

c) n = 20 p = .70

P(x < 12) = P(x=11) + P(x=10) + P(x=9) + ... + (Px=0) = .065 + .031 + .012 + .004 + .001 + .000 = .113

d) n = 20 p = .90

P(x < 16) = P(x=16) + P(x=15) + P(x=14) + ...+ P(x=0) = .090 + .032 + .009 + .002 + .000 = .133

e) n = 15 p = .40

P(4 < x < 9) = P(x=4) + P(x=5) + P(x=6) + P(x=7) + P(x=8) + P(x=9) =

.127 + .186 + .207 + .177 + .118 + .061 = .876

f) n = 10 p = .60

P(x > 7) = P(x=7) + P(x=8) + P(x=9) + P(x=10) = .215 + .122 + .040 + .006 = .382

5.7

a) n = 20 p = .70 q = .30 µ = n⋅p = 20(.70) = 14

σ = 2.4)30)(.70(.20 ==⋅⋅ qpn = 2.05

b) n = 70 p = .35 q = .65

µ = n⋅p = 70(.35) = 24.5


σ = 925.15)65)(.35(.70 ==⋅⋅ qpn = 3.99

c) n = 100 p = .50 q = .50 µ = n⋅p = 100(.50) = 50

σ = 25)50)(.50(.100 ==⋅⋅ qpn = 5

5.8 a) n = 6 p = .70 x Prob 0 .001 1 .010 2 .060 3 .185 4 .324 5 .303 6 .118

b) n = 20 p = .50 x Prob 0 .000 1 .000 2 .000 3 .001 4 .005 5 .015 6 .037 7 .074


8 .120 9 .160 10 .176 11 .160 12 .120 13 .074 14 .037 15 .015 16 .005 17 .001 18 .000 19 .000 20 .000

c) n = 8 p = .80 x Prob 0 .000 1 .000 2 .001 3 .009 4 .046 5 .147 6 .294 7 .336 8 .168


5.9 a) n = 20 p = .78 x = 14 20C14 (.78)14(.22)6 = 38,760(.030855)(.00011338) = .1356 b) n = 20 p = .75 x = 20 20C20 (.75)20(.25)0 = (1)(.0031712)(1) = .0032 c) n = 20 p = .70 x < 12 Use table A.2: P(x=0) + P(x=1) + . . . + P(x=11)= .000 + .000 + .000 + .000 + .000 + .000 + .000 + .001 + .004 + .012 + .031 + .065 = .113 5.10 n = 16 p = .40 P(x > 9): from Table A.2: x Prob 9 .084 10 .039 11 .014 12 .004 13 .001 .142


P(3 < x < 6): x Prob 3 .047 4 .101 5 .162 6 .198 .508 n = 13 p = .88 P(x = 10) = 13C10(.88)10(.12)3 = 286(.278500976)(.001728) = .1376 P(x = 13) = 13C13(.88)13(.12)0 = (1)(.1897906171)(1) = .1898 Expected Value = µ = n p = 13(.88) = 11.44 5.11 n = 25 P = .60 a) x > 15 P(x > 15) = P(x = 15) + P(x = 16) + · · · + P(x = 25) Using Table A.2 n = 25, p = .80 x Prob 15 .161 16 .151 17 .120 18 .080 19 .044 20 .020 21 .007 22 .002 .585 b) x > 20 from a): P(x > 20) = P(x = 21) + P(x = 22) + P(x = 23) + P(x = 24) + P(x = 25) = .007 + .002 + .000 + .000 + .000 = .009


c) P(x < 10) from Table A.2, x Prob. 9 .009 8 .003 7 .001 <6 .000 .013 5.12

The highest probability values are for x = 15, 16, 14, 17, 13, 18, and 12. The expected value is 25(.60) = 15. The standard deviation is 2.45. 15 + 2(2.45) = 15 + 4.90 gives a range that goes from 10.10 to 19.90. From table A.2, the sum of the probabilities of the values in this range (11 through 19) is .936 or 93.6% of the values which compares quite favorably with the 95% suggested by the empirical rule. 5.13 n = 15 p = .20 a) P(x = 5) = 15C5(.20)5(.80)10 = 3003(.00032)(.1073742) = .1032 b) P(x > 9): Using Table A.2 P(x = 10) + P(x = 11) + . . . + P(x = 15) =


.000 + .000 + . . . + .000 = .000 c) P(x = 0) = 15C0(.20)0(.80)15 = (1)(1)(.035184) = .0352 d) P(4 < x < 7): Using Table A.2 P(x = 4) + P(x = 5) + P(x = 6) + P(x = 7) = .188 + .103 + .043 + .014 = .348 e)

5.14 n = 18 a) p =.30 µ = 18(.30) = 5.4 p = .34 µ = 18(.34) = 6.12 b) P(x > 8) n = 18 p = .30 from Table A.2 x Prob 8 .081 9 .039 10 .015 11 .005 12 .001 .141


c) n = 18 p = .34 P(2 < x < 4) = P(x = 2) + P(x = 3) + P(x = 4) = 18C2(.34)2(.66)16 + 18C3(.34)3(.66)15 + 18C4(.34)4(.66)14 = .0229 + .0630 + .1217 = .2076 d) n = 18 p = .30 x = 0 18C0(.30)0(.70)18 = .00163 n = 18 p = .34 x = 0 18C0(.34)0(.66)18 = .00056 The probability that none are in the $500,000 to $1,000,000 is higher because there is a smaller percentage in that category which is closer to zero. 5.15 a) Prob(x=5λ = 2.3)= (2.35)(e-2.3) = (64.36343)(.1002588) = .0538 5! (120) b) Prob(x=2λ = 3.9) = (3.92)(e-3.9) = (15.21)(.02024) = .1539 2! (2) c) Prob(x < 3λ = 4.1) = Prob(x=3) + Prob(x=2) + Prob(x=1) + Prob(x=0) = (4.13)(e-4.1) = (68.921)(.016574) = .1904 3! 6 + (4.12)(e-4.1) = (16.81)(.016573) = .1393 2! 2 + (4.11)(e-4.1) = (4.1)(.016573) = .0679 1! 1


+ (4.10)(e-4.1) = (1)(.016573) = .0166 0! 1 .1904 + .1393 + .0679 + .0166 = .4142 d) Prob(x=0λ = 2.7) = (2.70)(e-2.7) = (1)(.0672) = .0672 0! 1 e) Prob(x=1 λ = 5.4)= (5.41)(e-5.4) = (5.4)(.0045) = .0244 1! 1 f) Prob(4 < x < 8 λ = 4.4) = Prob(x=5 λ = 4.4) + Prob(x=6 λ = 4.4) + Prob(x=7 λ = 4.4)= (4.45)(e-4.4) + (4.46)(e-4.4) + (4.47)(e-4.4) = 5! 6! 7! (1649.162)(.012277) + (7256.314)(.012277) + (31927.781)(.012277) 120 720 5040 = .1687 + .1237 + .0778 = .3702 5.16 a) Prob(x=6 λ = 3.8) = .0936 b) Prob(x>7 λ = 2.9): x Prob 8 .0068 9 .0022 10 .0006 11 .0002 12 .0000 x > 7 .0098


c) Prob(3 < x < 9 λ = 4.2)= x Prob 3 .1852 4 .1944 5 .1633 6 .1143 7 .0686 8 .0360 9 .0168 3 < x < 9 .7786 d) Prob(x=0 λ = 1.9) = .1496 e) Prob(x < 6 λ = 2.9)= x Prob 0 .0050 1 .1596 2 .2314 3 .2237 4 .1622 5 .0940 6 .0455 x < 6 .9214 f) Prob(5 < x < 8 λ = 5.7) = x Prob 6 .1594 7 .1298 8 .0925 5 < x < 8 .3817

5.17 a) λ = 6.3 mean = 6.3 Standard deviation = 3.6 = 2.51 x Prob 0 .0018 1 .0116 2 .0364 3 .0765 4 .1205 5 .1519 6 .1595


7 .1435 8 .1130 9 .0791 10 .0498 11 .0285 12 .0150 13 .0073 14 .0033 15 .0014 16 .0005 17 .0002 18 .0001 19 .0000

b) λ = 1.3 mean = 1.3 standard deviation = 3.1 = 1.14 x Prob 0 .2725 1 .3542 2 .2303 3 .0998 4 .0324 5 .0084 6 .0018 7 .0003 8 .0001 9 .0000


c) λ = 8.9 mean = 8.9 standard deviation = 9.8 = 2.98 x Prob 0 .0001 1 .0012 2 .0054 3 .0160 4 .0357 5 .0635 6 .0941 7 .1197 8 .1332 9 .1317 10 .1172 11 .0948 12 .0703 13 .0481 14 .0306 15 .0182 16 .0101 17 .0053 18 .0026 19 .0012 20 .0005 21 .0002 22 .0001


d) λ = 0.6 mean = 0.6 standard deviation = 6.0 = .775 x Prob 0 .5488 1 .3293 2 .0988 3 .0198 4 .0030 5 .0004 6 .0000

5.18 λ = 2.84 minutes a) Prob(x=6 λ = 2.8) from Table A.3 .0407


b) Prob(x=0 λ = 2.8) = from Table A.3 .0608 c) Unable to meet demand if x > 44 minutes: x Prob. 5 .0872 6 .0407 7 .0163 8 .0057 9 .0018 10 .0005 11 .0001 x > 4 .1523 .1523 probability of being unable to meet the demand. Probability of meeting the demand = 1 - (.1523) = .8477 15.23% of the time a second window will need to be opened.

d) λ = 2.8 arrivals4 minutes Prob(x=3) arrivals2 minutes = ?? Lambda must be changed to the same interval (½ the size) New lambda=1.4 arrivals2 minutes Prob(x=3) λ=1.4) = from Table A.3 = .1128 Prob(x > 5 8 minutes) = ?? Lambda must be changed to the same interval(twice the size): New lambda= 5.6 arrivals8 minutes Prob(x > 5 λ = 5.6):


From Table A.3: x Prob. 5 .1697 6 .1584 7 .1267 8 .0887 9 .0552 10 .0309 11 .0157 12 .0073 13 .0032 14 .0013 15 .0005 16 .0002 17 .0001 x > 5 .6579 5.19 λ = Σx/n = 126/36 = 3.5 Using Table A.3 a) P(x = 0) = .0302 b) P(x > 6) = P(x = 6) + P(x = 7) + . . . = .0771 + .0385 + .0169 + .0066 + .0023 + .0007 + .0002 + .0001 = .1424 c) P(x < 4 10 minutes) λ = 7.0 10 minutes P(x < 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) = .0009 + .0064 + .0223 + .0521 = .0817 d) P(3 < x < 6 10 minutes) λ = 7.0 10 minutes P(3 < x < 6) = P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) = .0521 + .0912 + .1277 + .1490 = .42


e) P(x = 8 15 minutes) λ = 10.5 15 minutes

P(x = 8 15 minutes) = !8

))(5.10(

!

5.108 −−

=⋅ e

X

eX λλ = .1009

5.20 λ = 5.6 days3 weeks a) Prob(x=0 λ = 5.6): from Table A.3 = .0037 b) Prob(x=6 λ = 5.6): from Table A.3 = .1584 c) Prob(x > 15 λ = 5.6): x Prob. 15 .0005 16 .0002 17 .0001 x > 15 .0008

Because this probability is so low, if it actually occurred, the researcher would actually have to question the Lambda value as too low for this period.

5.21 λ = 0.6 trips 1 year a) Prob(x=0 λ = 0.6): from Table A.3 = .5488 b) Prob(x=1 λ = 0.6): from Table A.3 = .3293 c) Prob(x > 2 λ = 0.6):


from Table A.3 x Prob. 2 .0988 3 .0198 4 .0030 5 .0004 6 .0000 x > 2 .1220 d) Prob(x < 3 3 year period): The interval length has been increased (3 times)

New Lambda = λ = 1.8 trips3 years

Prob(x < 3 λ = 1.8): from Table A.3 x Prob. 0 .1653 1 .2975 2 .2678 3 .1607 x < 3 .8913 e) Prob(x=4 6 years): The interval has been increased (6 times)

New Lambda = λ = 3.6 trips6 years

Prob(x=4 λ = 3.6): from Table A.3 = .1912 5.22 λ = 1.2 collisions4 months a) Prob(x=0 λ = 1.2): from Table A.3 = .3012 b) Prob(x=2 2 months):


The interval has been decreased (by ½) New Lambda = λ = 0.6 collisions2 months Prob(x=2 λ = 0.6): from Table A.3 = .0988 c) Prob (x < 1 collision6 months): The interval length has been increased (by 1.5) New Lambda = λ = 1.8 collisions6 months Prob(x < 1 λ = 1.8): from Table A.3 x Prob. 0 .1653 1 .2975 x < 1 .4628

The result is likely to happen almost half the time (46.26%). Ship channel and weather conditions are about normal for this period. Safety awareness is about normal for this period. There is no compelling reason to reject the lambda value of 0.6 collisions per 4 months based on an outcome of 0 or 1 collisions per 6 months.

5.23 λ = 1.2 penscarton a) Prob(x=0 λ = 1.2): from Table A.3 = .3012 b) Prob(x > 8 λ = 1.2): from Table A.3 = .0000 c) Prob(x > 3 λ = 1.2): from Table A.3 x Prob. 4 .0260 5 .0062


6 .0012 7 .0002 8 .0000 x > 3 .0336 5.24 n = 100,000 p = .00004 Prob(x > 7 n = 100,000 p = .00004): λ = µ = n⋅p = 100,000(.00004) = 4.0

Since n > 20 and n⋅p < 7, the Poisson approximation to this binomial problem is close enough.

Prob(x > 7 λ = 4): Using Table A.3 x Prob. 7 .0595 8 .0298 9 .0132 10 .0053 11 .0019 12 .0006 13 .0002 14 .0001 x > 7 .1106 Prob(x>10 λ = 4): Using Table A.3 x Prob. 11 .0019 12 .0006 13 .0002 14 .0001 x > 10 .0028

Since getting more than 10 is a rare occurrence, this particular geographic region appears to have a higher average rate than other regions. An investigation of particular characteristics of this region might be warranted.


5.25 p = .009 n = 200 Use the Poisson Distribution: λ = n⋅p = 200(.009) = 1.8 P(x > 6) from Table A.3 = P(x = 6) + P(x = 7) + P(x = 8) + P(x = 9) + . . . = .0078 + .0020 + .0005 + .0001 = .0104 P(x > 10) = .0000 P(x = 0) = .1653 P(x < 5) = P(x = 0) + P(x = 1) + P(x = 2) + P( x = 3) + P(x = 4) = .1653 + .2975 + .2678 + .1607 + .0723 = .9636 5.26 n = 300, p = .01, λ = n(p) = 300(.01) = 3 a) Prob(x = 5): Using λ = 3 and Table A.3 = .1008 b) Prob (x < 4) = Prob.(x = 0) + Prob.(x = 1) + Prob.(x = 2) + Prob.(x = 3) = .0498 + .1494 + .2240 + .2240 = .6472 c) The expected number = µ = λ = 3 5.27 a) Prob(x = 3 N = 11, A = 8, n = 4)

330

)3)(56(

411

1338 =⋅C

CC = .5091

b) Prob(x < 2)N = 15, A = 5, n = 6) Prob(x = 1) + Prob (x = 0) =

615

51015

C

CC ⋅+

615

61005

C

CC ⋅ =

5005

)210)(1(

5005

)252)(5( +


.2517 + .0420 = .2937 c) Prob(x=0 N = 9, A = 2, n = 3)

84

)35)(1(

39

3702 =⋅C

CC = .4167

d) Prob(x > 4 N = 20, A = 5, n = 7) = Prob(x = 5) + Prob(x = 6) + Prob(x = 7) =

720

21555

C

CC ⋅+

720

11565

C

CC ⋅ +

720

01575

C

CC ⋅ =

77520

)105)(1( + 5C6 (impossible) + 5C7(impossible) = .0014

5.28 N = 19 n = 6 a) P(x = 1 private) A = 11

132,27

)56)(11(

619

58111 =⋅C

CC = .0227

b) P(x = 4 private)

132,27

)28)(330(

619

28411 =⋅C

CC = .3406

c) P(x = 6 private)

132,27

)1)(462(

619

08611 =⋅C

CC = .0170

d) P(x = 0 private)

132,27

)28)(1(

619

68011 =⋅C

CC = .0010


5.29 N = 17 A = 8 n = 4

a) P(x = 0) = 417

4908

C

CC ⋅ =

2380

)126)(1( = .0529

b) P(x = 4) = 417

0948

C

CC ⋅ =

2380

)1)(70( = .0294

c) P(x = 2 non computer) = 417

2829

C

CC ⋅ =

2380

)28)(36( = .4235

5.30 N = 20 A = 16 white N - A = 4 red n = 5

a) Prob(x = 4 white) = 520

14416

C

CC ⋅ =

15504

)4)(1820( = .4696

b) Prob(x = 4 red) = 520

11644

C

CC ⋅ =

15504

)16)(1(= .0010

c) Prob(x = 5 red) = 520

01654

C

CC ⋅ = .0000 because 4C5 is impossible to determine

The participant cannot draw 5 red beads if there are only 4 to draw from. 5.31 N = 10 n = 4 a) A = 3 x = 2

210

)21)(3(

410

2723 =⋅C

CC = .30

b) A = 5 x = 0

210

)5)(1(

410

4505 =⋅C

CC = .0238

c) A = 5 x = 3

210

)5)(10(

410

1535 =⋅C

CC = .2381


5.32 N = 16 A = 4 defective n = 3

a) Prob(x = 0) = 560

)220)(1(

316

31204 =⋅C

CC = .3929

b) Prob(x = 3) = 560

)1)(4(

316

01234 =⋅C

CC = .0071

c) Prob(x > 2) = Prob(x=2) + Prob(x=3) = 316

11224

C

CC ⋅ + .0071 (from part b.) =

560

)12)(6( + .0071 = .1286 + .0071 = .1357

d) Prob(x < 1) = Prob(x=1) + Prob(x=0) =

316

21214

C

CC ⋅ + .3929 (from part a.) =

560

)66)(4( + .3929 = .4714 + .3929 = .8643

5.33 N = 18 A = 11 Hispanic n = 5 Prob(x < 1) = Prob(1) + Prob(0) =

518

47111

C

CC ⋅+

518

57011

C

CC ⋅ =

8568

)21)(1(

8568

)35)(11( + = .0449 + .0025 = .0474

It is fairly unlikely that these results occur by chance. A researcher might investigate further the causes of this result. Were officers selected based on leadership, years of service, dedication, prejudice, or what?

5.34 a) Prob(x=4 n = 11 and p = .23) 11C4(.23)4(.77)7 = 330(.0028)(.1605) = .1482 b) Prob(x > 1n = 6 and p = .50) = 1 - Prob(x < 1) = 1 - Prob(x = 0) = 1-6C0(.50)0(.50)6 = 1-(1)(1)(.0156) = .9844


c) Prob(x > 7 n = 9 and p = .85) = Prob(x = 8) + Prob(x = 9) = 9C8(.85)8(.15)1 + 9C9(.85)9(.15)0 = (9)(.2725)(.15) + (1)(.2316)(1) = .3679 + .2316 = .5995 d) Prob(x < 3 n = 14 and p = .70) = Prob(x = 3) + Prob(x = 2) + Prob(x = 1) + Prob(x = 0) = 14C3(.70)3(.30)11 + 14C2(.70)2(.30)12 + 14C1(.70)1(.30)13 + 14C0(.70)0(.30)14 = (364)(.3430)(.00000177) + (91)(.49)(.000000047)= (14)(.70)(.00000016) + (1)(1)(.000000047) = .0002 + .0000 + .0000 + .0000 = .0002 5.35 a) Prob(x = 14 n = 20 and p = .60) = .124 b) Prob(x < 5 n = 10 and p =.30) = Prob(x = 4) + Prob(x = 3) + Prob(x = 2) + Prob(x = 1) + Prob(x=0) = x Prob. 0 .028 1 .121 2 .233 3 .267 4 .200 x < 5 .849 c) Prob(x > 12 n = 15 and p = .60) = Prob(x = 12) + Prob(x = 13) + Prob(x = 14) + Prob(x = 15) x Prob. 12 .063 13 .022


14 .005 15 .000 x > 12 .090 d) Prob(x > 20 n = 25 and p = .40) = Prob(x = 21) + Prob(x = 22) + Prob(x = 23) + Prob(x = 24) + Prob(x=25) = x Prob. 21 .000 22 .000 23 .000 24 .000 25 .000 x > 20 .000 5.36 a) Prob(x=4 λ = 1.25) (1.254)(e-1.25) = (2.4414)(.2865) = .0291 4! 24 b) Prob(x < 1 λ = 6.37) = Prob(x = 1) + Prob(x = 0) = (6.37)1(e-6.37) + (6.37)0(e-6.37) = (6.37)(.0017) + (1)(.0017) = 1! 0! 1 1 .0109 + .0017 = .0126 c) Prob(x > 5 λ = 2.4) = Prob(x = 6) + Prob(x = 7) + ... = (2.4)6(e-2.4) + (2.4)7(e-2.4) + (2.4)8(e-2.4) + (2.4)9(e-2.4) + (2.4)10(e-2.4) + ... 6! 7! 8! 9! 10! .0241 + .0083 + .0025 + .0007 + .0002 = .0358

for values x > 11 the probabilities are each .0000 when rounded off to 4 decimal places.


5.37 a) Prob(x = 3 λ = 1.8) = .1607 b) Prob(x < 5 λ = 3.3) = Prob(x = 4) + Prob(x = 3) + Prob(x = 2) + Prob(x = 1) + Prob(x = 0) = x Prob. 0 .0369 1 .1217 2 .2008 3 .2209 4 .1823 x < 5 .7626 c) Prob(x > 3 λ = 2.1) = x Prob. 3 .1890 4 .0992 5 .0417 6 .0146 7 .0044 8 .0011 9 .0003 10 .0001 11 .0000 x > 5 .3504 d) Prob(2 < x < 5 λ = 4.2) = Prob(x=3) + Prob(x=4) + Prob(x=5) = x Prob. 3 .1852 4 .1944 5 .1633 2 < x < 5 .5429 5.38 a) Prob(x = 3 N = 6, n = 4, A = 5) =

15

)1)(10(

46

1135 =⋅

C

CC = .6667


b) Prob(x < 1 N = 10, n = 3, A = 5) = Prob(x = 1) + Prob(x = 0) =

310

2515

C

CC ⋅+

310

3505

C

CC ⋅ =

120

)10)(1(

120

)10)(5( +

= .4167 + .0833 = .5000 c) Prob(x > 2 N = 13, n = 5, A = 3) = Prob(x=2) + Prob(x=3) Note: only 3 x's in population

513

31023

C

CC ⋅ +

513

21033

C

CC ⋅ =

1287

)45)(1(

1287

)120)(3( +

= .2797 + .0350 = .3147 5.39 n = 25 p = .20 retired from Table A.2: P(x = 7) = .111 P(x > 10): P(x = 10) + P(x = 11) + . . . + P(x = 25) = .012 + .004 + .001 = .017 Expected Value = µ = n⋅p = 25(.20) = 5 n = 20 p = .40 mutual funds P(x = 8) = .180 P(x < 6) = P(x = 0) + P(x = 1) + . . . + P(x = 5) = .000 + .000 + .003 +.012 + .035 + .075 = .125 P(x = 0) = .000 P(x > 12) = P(x = 12) + P(x = 13) + . . . + P(x = 20) = .035 + .015 + .005 + .001 = .056 x = 8 Expected Number = µ = n p = 20(.40) = 8


5.40 λ = 3.2 cars2 hours a) Prob(x=3) cars per 1 hour) = ?? The interval has been decreased by ½. The new λ = 1.6 cars1 hour. Prob(x = 3 λ = 1.6) = (from Table A.3) .1378 b) Prob(x = 0 cars per ½ hour) = ?? The interval has been decreased by ¼ the original amount. The new λ = 0.8 cars½ hour. Prob(x = 0 λ = 0.8) = (from Table A.3) .4493 c) Prob(x > 5 λ = 1.6) = (from Table A.3) x Prob. 5 .0176 6 .0047 7 .0011 8 .0002 .0236

Either a rare event occurred or perhaps the long-run average, λ, has changed (increased).

5.41 N = 32 A = 10 n = 12

a) P(x = 3) = 1232

922310

C

CC ⋅ =

840,792,225

)420,497)(120( = .2644

b) P(x = 6) = 1232

622610

C

CC ⋅ =

840,792,225

)613,74)(210( = .0694

c) P(x = 0) = 1232

1222010

C

CC ⋅ =

840,792,225

)646,646)(1( = .0029


d) A = 22

P(7 < x < 9) = 1232

510722

C

CC ⋅+

1232

410822

C

CC ⋅+

1232

310922

C

CC ⋅

= 840,792,225

)120)(420,497(

840,792,225

)210)(770,319(

840,792,225

)252)(544,170( ++

= .1903 + .2974 + .2644 = .7521 5.42 λ = 1.4 defects 1 lot If x > 3, buyer rejects If x < 3, buyer accepts Prob(x < 3 λ = 1.4) = (from Table A.3) x Prob. 0 .2466 1 .3452 2 .2417 3 .1128 x < 3 .9463 5.43 a) n = 20 and p = .25 The expected number = µ = n⋅p = (20)(.25) = 5.00 b) Prob(x < 1 n = 20 and p = .25) = Prob(x = 1) + Prob(x = 0) = 20C1(.25)1(.75)19 + 20C0(.25)0(.75)20 = (20)(.25)(.00423) + (1)(1)(.0032) = .0212 +. 0032 = .0244 Since the probability is so low, the population of your state may have a lower percentage of chronic heart conditions than those of other states. 5.44 a) Prob(x > 7 n = 10 and p = .70) = (from Table A.2): x Prob. 8 .233 9 .121 10 .028 x > 7 .382


Expected number = µ = n⋅p = 10(.70) = 7 b) Expected number = µ = n⋅p = 15 (1/3) = 5 Prob(x=0⋅ n = 15 and p = 1/3) = 15C0(�)0(�)15 = .0023 c) Prob(x = 7 n = 7 and p = .53) = 7C7(.53)7(.47)0 = .0117

Probably the 53% figure is too low for this population since the probability of this occurrence is so low (.0117).

5.45 n = 12 a.) Prob(x = 0 long hours): p = .20 12C0(.20)0(.80)12 = .0687 b.) Prob(x > 6) long hours): p = .20 Using Table A.2: .016 + .003 + .001 = .020 c) Prob(x = 5 good financing): p = .25, 12C5(.25)5(.75)7 = .1032 d.) p = .19 (good plan), expected number = µ = n(p) = 12(.19) = 2.28 5.46 n = 100,000 p = .000014 Worked as a Poisson: λ = n⋅p = 100,000(.000014) = 1.4 a) P(x = 5): from Table A.3 = .0111 b) P(x = 0):


from Table A.3 = .2466 c) P(x > 6): (from Table A.3) x Prob 7 .0005 8 .0001 .0006 5.47 Prob(x < 3) n = 8 and p = .60) = (from Table A.2) x Prob. 0 .001 1 .008 2 .041 3 .124 x < 3 .174

17.4% of the time in a sample of eight, three or fewer customers are walk-ins by chance. Other reasons for such a low number of walk-ins might be that she is retaining more old customers than before or perhaps a new competitor is attracting walk-ins away from her.

5.48 n = 25 p = .20 a) Prob(x = 8 n = 25 and p = .20) = (from Table A.2) .062 b) Prob(x > 10) n=25 and p = .20) = (from Table A.2) x Prob. 11 .004 12 .001 13 .000 x > 10 .005

c) Since such a result would only occur 0.5% of the time by chance, it is likely that the analyst's list was not representative of the entire state of Idaho or the 20% figure for the Idaho census is not correct.

5.49 λ = 0.6 flats2000 miles Prob(x = 0 λ = 0.6) = (from Table A.3) .5488


Prob(x > 3 λ = 0.6) = (from Table A.3) x Prob. 3 .0198 4 .0030 5 .0004 x > 3 .0232

One trip is independent of the other. Let F = flat tire and NF = no flat tire P(NF1 _ NF2) = P(NF1) ⋅ P(NF2) P(NF) = .5488 P(NF1 _ NF2) = (.5488)(.5488) = .3012 5.50 N = 25 n = 8 a) P(x = 1 in NY) A = 4

825

72114

C

CC ⋅=

575,081,1

)280,116)(4( = .4300

b) P(x = 4 in top 10) A = 10

575,081,1

)1365(210(

825

415410 =⋅C

CC = .2650

c) P(x = 0 in California) A = 4

575,081,1

)490,203)(1(

825

82104 =⋅C

CC = .1881

d) P(x = 3 with M) A = 3

575,081,1

)334,26)(1(

825

52233 =⋅C

CC = .0243

5.51 N = 24 n = 6 A = 8


a) P(x = 6) = 596,134

)1)(28(

624

01668 =⋅C

CC = .0002

b) P(x = 0) = 596,134

)8008)(1(

624

61608 =⋅C

CC = .0595

d) A = 16 East Side

P(x = 3) = 596,134

)56)(560(

624

38316 =⋅C

CC = .2330

5.52 n = 25 p = .20 Expected Value = µ = n⋅p = 25(.20) = 5

µ = 25(.20) = 5

σ = )80)(.20(.25=⋅⋅ qpn = 2

P(x > 12) = (from Table A.2) x Prob 13 .0000

The values for x > 12 are so far away from the expected value that they are very unlikely to occur.

P(x = 14) = 25C14(.20)14(.80)11 = .000063 which is very unlikely. If this value (x = 14) actually occurred, one would doubt the validity of the p = .20 figure or one would have experienced a very rare event.


5.53 λ = 2.4 calls1 minute a) Prob(x = 0 λ = 2.4) = (from Table A.3) .0907 b) Can handle x < 5 calls Cannot handle x > 5 calls Prob(x > 5 λ = 2.4) = (from Table A.3) x Prob. 6 .0241 7 .0083 8 .0025 9 .0007 10 .0002 11 .0000 x > 5 .0358 c) Prob(x = 3 calls 2 minutes) The interval has been increased 2 times. New Lambda = λ = 4.8 calls2 minutes. from Table A.3: .1517 d) Prob(x < 1 calls15 seconds): The interval has been decreased by ¼. New Lambda = λ = 0.6 calls15 seconds. Prob(x < 1 λ = 0.6) = (from Table A.3) Prob(x = 1) = .3293 Prob(x = 0) = .5488 Prob(x < 1) = .8781


5.54 n = 160 p = .01 Working this problem as a Poisson problem: a) Expected number = µ = n(p) = 160(.01) = 1.6 b) P(x > 8): Using Table A.3: x Prob. 8 .0002 9 .0000 .0002 c) P(2 < x < 6): Using Table A.3: x Prob. 2 .2584 3 .1378 4 .0551 5 .0176 6 .0047 P(2 < x < 6) .4736 5.55 p = .005 n = 1,000 λ = n⋅p = (1,000)(.005) = 5 a) P(x < 4) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) = .0067 + .0337 + .0842 + .1404 = .265 b) P(x > 10) = P(x = 11) + P(x = 12) + . . . = .0082 + .0034 + .0013 + .0005 + .0002 = .0136 c) P(x = 0) = .0067


5.56 n = 8 P = .36 x = 0 Women 8C0(.36)0(.64)8 = (1)(1)(.0281475) = .0281 It is unlikely that a company would randomly hire 8 physicians from the U.S. pool and none of them would be female. If this actually happened, the figures might be used as evidence in a lawsuit. 5.57 N = 32 a) n = 5 x = 3 A = 10

376,201

)231)(120(

532

222310 =⋅C

CC = .1377

b) n = 8 x < 2 A = 6

832

82606

C

CC ⋅ +

832

72616

C

CC ⋅ +

832

62626

C

CC ⋅ =

300,518,10

)760,38)(15(

300,518,10

)800,657)(6(

300,518,10

)275,562,1)(1( ++ = .1485 + .3752 + .0553 = .5790

c) n = 5 x = 2 p = 3/26 = .1154 5C2(.1154)2(.8846)3 = (10)(.013317)(.692215) = .0922 5.58 N = 14 n = 4 a) P(x = 4 N = 14, n = 4, A = 10 Northside)

1001

)1((210(

414

04410 =⋅C

CC = .2098

b) P(x = 4 N = 14, n = 4, A = 4 West)

1001

)1)(1(

414

01044 =⋅C

CC = .0010

c) P(x = 2 N = 14, n = 4, A = 4 West)


1001

)45)(6(

414

21024 =⋅C

CC = .2697

5.59 a) λ = 3.841,000

P(x = 0) = !0

84.3 84.30 −⋅ e = .0215

b) λ = 7.682,000

P(x = 6) = 720

)000461975)(.258.195,205(

!6

68.7 68.76

=⋅ −e = .1317

c) λ = 1.61,000 and λ = 4.83,000 from Table A.3: P(x < 7) = P(x = 0) + P(x = 1) + . . . + P(x = 6) = .0082 + .0395 + .0948 + .1517 + .1820 + .1747 + .1398 = .7907 5.60 This is a binomial distribution with n = 15 and p = .36. µ = n⋅p = 15(.36) = 5.4

σ = )64)(.36(.15 = 1.86

The most likely values are near the mean, 5.4. Note from the printout that the most probable values are at x = 5 and x = 6 which are near the mean. 5.61 This printout contains the probabilities for various values of x from zero to eleven from a

Poisson distribution with λ = 2.78. Note that the highest probabilities are at x = 2 and x = 3 which are near the mean. The probability is slightly higher at x = 2 than at x = 3 even though x = 3 is nearer to the mean because of the “piling up” effect of x = 0.

5.62 This is a binomial distribution with n = 22 and p = .64. The mean is n⋅p = 22(.64) = 14.08 and the standard deviation is:

σ = )36)(.64(.22=⋅⋅ qpn = 2.25


The x value with the highest peak on the graph is at x = 14 followed by x = 15 and x = 13 which are nearest to the mean. 5.63 This is the graph of a Poisson Distribution with λ = 1.784. Note the high probabilities at x = 1 and x = 2 which are nearest to the mean. Note also that the probabilities for values of x > 8 are near to zero because they are so far away from the mean or expected value.

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05 ch ken black solution