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
 
  P(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) (1-p)
 
px (1-p)n-x
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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
12.
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3 out of
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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
13.
award:
3 out of
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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

29.
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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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3 out of
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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
38.
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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
 
46.
award:
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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
47.
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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
48.
award:
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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
53.
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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
57.
award:
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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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award:
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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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