probability distribution
Econ2300
assignment: Ch5 Quiz
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Exercise 5.12 METHODS AND APPLICATIONS
Suppose that the probability distribution of a random variable x can be described by the formula
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Order Paper NowP(x) = x
________________________________________
15
for each of the values x = 1, 2, 3, 4, and 5. For example, then, P(x = 2) = p(2) =2/15.
(a) Write out the probability distribution of x. (Write all fractions in reduced form.)
x 1 2 3 4 5
P(x) ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________
________________________________________
(b) Show that the probability distribution of x satisfies the properties of a discrete probability distribution.(Round other answers to the nearest whole number. Leave no cells blank – be certain to enter “0” wherever required.)
P(x) ≥ for each value of x.
(c) Calculate the mean of x. (Round your answer to 3 decimal places.)
µx
(d) Calculate the variance, σ2x , and the standard deviation, σx. (Round your answer σx2 in to 3 decimal places and round answer σx in to 4 decimal places.)
σx2
σx
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Exercise 5.23 METHODS AND APPLICATIONS
Suppose that x is a binomial random variable with n = 5, p = 0.3, and q = 0.7.
(b) For each value of x, calculate p(x), and graph the binomial distribution. (Round final answers to 5 decimal places.)
p(0) = , p(1) = , p(2) = , p (3) = ,
p(4) = , p(5) =
(c) Find P(x = 3). (Round final answer to 5 decimal places.)
P(x=3)
(d) Find P(x ≤ 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)
P(x ≤ 3)
(e) Find P(x < 3). (Do not round intermediate calculations. Round final answer to 5 decimal places.)
P(x < 3) = P(x ≤ 2)
(f) Find P(x ≥ 4). (Do not round intermediate calculations. Round final answer to 5 decimal places.)
P(x ≥ 4)
(g) Find P(x > 2). (Do not round intermediate calculations. Round final answer to 5 decimal places.)
P(x > 2)
(h) Use the probabilities you computed in part b to calculate the mean, μx, the variance, σ 2x, and the standard deviation, σx, of this binomial distribution. Show that the formulas for μx , σ 2x, and σx given in this section give the same results. (Do not round intermediate calculations. Round final answers to µx and σ 2x in to 2 decimal places, and σx in to 6 decimal places.)
µx
σ2x
σx
(i) Calculate the interval [μx ± 2σx]. Use the probabilities of part b to find the probability that x will be in this interval. Hint: When calculating probability, round up the lower interval to next whole number and round down the upper interval to previous whole number. (Round your answers to 5 decimal places. A negative sign should be used instead of parentheses.)
The interval is [ , ].
P( ≤ x ≤ ) =
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MC Qu. 14 The mean of the binomial distribution is equ…
The mean of the binomial distribution is equal to:
p
np
(n) (p) (1p)
px (1p)nx
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MC Qu. 25 A fair die is rolled 10 times. What is the p…
A fair die is rolled 10 times. What is the probability that an odd number (1, 3, or 5) will occur less than 3 times?
.1550
.8450
.0547
.7752
.1172
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MC Qu. 31 If n = 20 and p = .4, then the mean of the b…
If n = 20 and p = .4, then the mean of the binomial distribution is
.4
4.8
8
12
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MC Qu. 36 The probability that a given computer chip w…
The probability that a given computer chip will fail is 0.02. Find the probability that of 5 delivered chips, exactly 2 will fail.
.9039
.0000
.0922
.0038
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MC Qu. 38 In the most recent election, 19% of all elig…
In the most recent election, 19% of all eligible college students voted. If a random sample of 20 students were surveyed:
Find the probability that exactly half voted in the election.
.4997
.0014
.0148
.0000
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MC Qu. 39 In the most recent election, 19% of all elig…
In the most recent election, 19% of all eligible college students voted. If a random sample of 20 students were surveyed:
Find the probability that none of the students voted.
.0148
.4997
.0014
.0000
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MC Qu. 55 For a random variable X, the mean value of t…
For a random variable X, the mean value of the squared deviations of its values from their expected value is called its
Standard Deviation
Mean
Probability
Variance
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MC Qu. 62 If the probability distribution of X is:&nbs…
If the probability distribution of X is:
What is the expected value of X?
2.25
2.24
1.0
5.0
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MC Qu. 63 If the probability distribution of X is:&nbs…
If the probability distribution of X is:
What is the variance of X?
5.0
→
1.0
2.25
2.24
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MC Qu. 66 A vaccine is 95 percent effective. What is t…
A vaccine is 95 percent effective. What is the probability that it is not effective for, more than one out of 20 individuals?
.3774
.7359
→
.2641
.3585
P(X ≥ 2) = 1 – [P(X = 0) + p(X = 1)]
P(X ≥ 2) = 1 – [(.3585) + (.3774)] = .2641
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MC Qu. 67 If the probability of a success on a single …
If the probability of a success on a single trial is .2, what is the probability of obtaining 3 successes in 10 trials if the number of successes is binomial?
.1074
.2013
.5033
.0031
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MC Qu. 78 For a binomial process, the probability of s…
For a binomial process, the probability of success is 40% and the number of trials is 5.
Find the expected value.
5.0
1.1
1.2
2.0
E[X] = (5) (.40) = 2
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MC Qu. 79 For a binomial process, the probability of s…
For a binomial process, the probability of success is 40% and the number of trials is 5.
Find the variance.
1.1
→
1.2
5.0
2.0
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MC Qu. 80 For a binomial process, the probability of s…
For a binomial process, the probability of success is 40% and the number of trials is 5.
Find the standard deviation.
1.1
5.0
2.0
1.2
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MC Qu. 82 For a binomial process, the probability of s…
For a binomial process, the probability of success is 40% and the number of trials is 5.
Find P(X > 4).
.0102
.0778
.0870
.3370
P(X = 5) = (.4)5 = .0102
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MC Qu. 92 If X has the probability distribution%…
If X has the probability distribution
compute the expected value of X.
0.5
0.7
1.0
0.3
E[X] = 1(.2) + 0(.3) + 1(.5) = .3
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MC Qu. 93 If X has the probability distribution%…
If X has the probability distribution
compute the expected value of X.
1.3
2.4
1.0
1.8
E[X] = (2) (.2) + (1) (.2) + (1) (.2) + (2) (.2) + (9) (.2) = 1.8
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MC Qu. 95 X has the following probability distribution…
X has the following probability distribution P(X):
Compute the variance value of X.
1.58
.625
→
.850
.955
E[X] = (1) (.1) + (2) (.5) + (3) (.2) + (4) (.2) = 2.5
= (1 – 2.5)2 (.1) + (2 – 2.5)2 (.5) + (3 – 2.5)2 (.2) + (4 – 2.5)2 (.2) = .850
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MC Qu. 99 Consider the experiment of tossing a fair co…
Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads).
Determine the expected number of heads.
1.1
1.5
1.0
2.0
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MC Qu. 100 Consider the experiment of tossing a fair co…
Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads).
What is the variance for this distribution?
→
0.75
0.87
1.22
1.5
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MC Qu. 102 Consider the experiment of tossing a fair co…
Consider the experiment of tossing a fair coin three times and observing the number of heads that result (X = number of heads).
If you were asked to play a game in which you tossed a fair coin three times and were given $2 for every head you threw, how much would you expect to win on average?
$6
→
$3
$9
$2
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MC Qu. 104 According to data from the state blood progr…
According to data from the state blood program, 40% of all individuals have group A blood. If six (6) individuals give blood, find the probability that None of the individuals has group A blood?
.0467
.4000
.0041
.0410
View Hint #1
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MC Qu. 105 According to data from the state blood progr…
According to data from the state blood program, 40% of all individuals have group A blood. If six (6) individuals give blood, find the probability that Exactly three of the individuals has group A blood?
.4000
.2765
.5875
.0041
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MC Qu. 106 According to data from the state blood progr…
According to data from the state blood program, 40% of all individuals have group A blood. If six (6) individuals give blood, find the probability that At least 3 of the individuals have group A blood.
.4557
.8208
.1792
.5443
P(x ≥ 3) = P(x = 3) + p(x = 4) + p(x = 5) + p(x = 6) = .4557
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MC Qu. 113 An important part of the customer service re…
An important part of the customer service responsibilities of a cable company relates to the speed with which trouble in service can be repaired. Historically, the data show that the likelihood is 0.75 that troubles in a residential service can be repaired on the same day. For the first five troubles reported on a given day, what is the probability that fewer than two troubles will be repaired on the same day?
.0010
.0146
→
.0156
.6328
P(x < 2) = P(x = 0) + P(x = 1) = .0156
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MC Qu. 116 The Post Office has established a record in …
The Post Office has established a record in a major Midwestern city for delivering 90% of its local mail the next working day. If you mail eight local letters, what is the probability that all of them will be delivered the next day.
1.0
.5695
→
.4305
.8131
P(x = 8) = .4305
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MC Qu. 117 The Post Office has established a record in …
The Post Office has established a record in a major Midwestern city for delivering 90% of its local mail the next working day. Of the eight, what is the average number you expect to be delivered the next day?
4.0
2.7
3.6
7.2
= np = (8) (.9) = 7.2
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MC Qu. 118 The Post Office has established a record in …
The Post Office has established a record in a major Midwestern city for delivering 90% of its local mail the next working day. Calculate the standard deviation of the number delivered when 8 local letters are mailed.
.72
→
.85
2.83
2.68
Σ = = = = .85
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MC Qu. 124 A large disaster cleaning company estimates …
A large disaster cleaning company estimates that 30% of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that is has contracted calculate the mean number of jobs completed within the bid time.
2.0
5.6
4.0
2.4
= np = 8(.3) = 2.4
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MC Qu. 125 A large disaster cleaning company estimates …
A large disaster cleaning company estimates that 30% of the jobs it bids on are finished within the bid time. Looking at a random sample of 8 jobs that is has contracted find the probability that x (number of jobs finished on time) is within one standard deviation of the mean.
→
.5506
.6867
.8844
.7483
σ = √npq = 1.3. P(µ+/σ) = P(2.4+/1.3) = P(1.1≤X≤3.7) = P(2≤X≤3) = 0.2965+0.2541 = 0.5506
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