Question 1

Your company's internal auditor believes that 10 percent of the company's invoices contain errors.To check this theory,20 invoices are randomly selected,and 5 are found to have errors.What is the probability that of the 20 invoices selected,5 or more would contain errors if the theory is valid?
A).0433
B).0319
C).9567
D).8660

Accepted Answer

The probability of finding 5 or more errors in 20 invoices, given a 10% error rate, can be calculated using the binomial distribution. The probability of exactly 5 errors is calculated using the binomial probability formula, and then the cumulative probability for 5 or more errors is found. The correct cumulative probability is approximately 0.0433.

Question 2

The Post Office has established a record in a major midwestern city for delivering 90 percent 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?
A)1.0
B).4305
C).8131
D).5695

Accepted Answer

The probability of one letter being delivered the next day is 0.9. Therefore, the probability of all eight letters being delivered the next day is 0.9^8, which is approximately 0.4305.

Question 3

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?
A).6328
B).0010
C).0156
D).0146

Accepted Answer

The given probability of troubles being repaired on the same day is p = 0.75
Let X be the number of troubles repaired on the same day out of the first 5 troubles reported.
X follows a binomial distribution with n = 5 and p = 0.75
We need to find P(X < 2)
P(X < 2) = P(X = 0) + P(X = 1)
P(X = 0) = (5 choose 0) * (0.75)^0 * (0.25)^5 = 0.001
P(X = 1) = (5 choose 1) * (0.75)^1 * (0.25)^4 = 0.014
Therefore, P(X < 2) = 0.001 + 0.014 = 0.0156, which is closest to choice C.

Question 4

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 at least three troubles will be repaired on the same day?
A).1035
B).0376
C).9624
D).8965

Accepted Answer

The probability of at least three troubles being repaired on the same day can be calculated using the binomial probability formula. The probability of 3, 4, or 5 repairs is calculated and summed: P(X ≥ 3) = P(X=3) + P(X=4) + P(X=5) = 0.2637 + 0.3955 + 0.2373 = 0.8965.

Question 5

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the probability that none of the individuals has group A blood.
A).0041
B).0410
C).4000
D).0467

Accepted Answer

P(x = 0) = .0467

Question 6

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?
A)$3
B)$2
C)$6
D)$9

Accepted Answer

Expected return = 0+ ($2)(1)+ ($2)(2)+ ($2)(3)= 3.0

Question 7

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 all five will be repaired on the same day?
A).0010
B).6328
C).9990
D).2373

Accepted Answer

The probability of one trouble being repaired on the same day is 0.75, and we assume that these events are independent. Therefore, the probability that all five reported troubles are repaired on the same day is:

0.75 x 0.75 x 0.75 x 0.75 x 0.75 = 0.2373

So, the best choice is D - 0.2373.

Question 8

A pharmaceutical company has determined that if a new cholesterol-reducing drug is manufactured (introduced to the market),the following probability distribution will describe this drug's contribution to the company's profits during the next six months. $\begin{array} { c c } 
\text { Profit Contribution } & \text { Probability of Profit Contribution } \
\hline - \$ 30,000 & .20 \
\$ 50,000 & .50 \
\$ 200,000 & .30
\end{array}$
The company management has decided to market this product if the expected contribution to profit for the next six months is more than $90,000.Based on the information given above,should the company begin manufacturing the new drug? Explain your answer.
A)Yes,begin manufacturing.
B)No,don't begin manufacturing.
$\mu _ { x }$

Accepted Answer

The expected profit contribution is calculated as: (-$30,000 * 0.20) + ($50,000 * 0.50) + ($200,000 * 0.30) = -$6,000 + $25,000 + $60,000 = $79,000. Since $79,000 is less than $90,000, the company should not begin manufacturing.

Question 9

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day.When there are 8 local letters mailed,what is the probability that the number delivered will be within 2 standard deviations of the mean?
A).9950
B).9619
C).8131
D).9996

Accepted Answer

To find the probability that the number of letters delivered is within 2 standard deviations of the mean, we need to first find the mean and standard deviation. 
If 90% of local mail is delivered the next working day, then the mean number of letters delivered is:
0.9 * 8 = 7.2
The standard deviation can be calculated using the formula:
sqrt(npq) = sqrt(8*0.9*0.1) = 0.949
To find the number of letters delivered within 2 standard deviations of the mean, we need to calculate the upper and lower limits:
Upper limit = mean + 2 * standard deviation = 7.2 + 2*0.949 = 9.098
Lower limit = mean - 2 * standard deviation = 7.2 - 2*0.949 = 5.302
Now we can use the normal distribution to find the probability that the number of letters delivered is between 5.302 and 9.098:
z = (x - mean)/standard deviation
For x = 5.302: z = (5.302 - 7.2)/0.949 = -1.9987
For x = 9.098: z = (9.098 - 7.2)/0.949 = 1.9910
Using a z-table, we can find that the probability of z being between -1.9987 and 1.9910 is 0.9619, which is answer choice B.

Question 10

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the mean number of individuals having group A blood.
A)1.2
B)1.55
C)1.44
D)2.4

Accepted Answer

μ_x = np = (6)(.4) = 2.4

Question 11

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 standard deviation for this distribution?
A)1.5
B)1.22
C)0.75
D)0.87

Accepted Answer

The distribution of the number of heads in three tosses of a fair coin is a binomial distribution with parameters n=3 and p=0.5. The formula for the standard deviation of a binomial distribution is:

SD = sqrt(np(1-p))

Plugging in n=3 and p=0.5, we get:

SD = sqrt(3*0.5*(1-0.5)) = 0.866

So the standard deviation is 0.87, which is closest to choice D.

Question 12

A lawyer believes that the probability is .3 that she can win a discrimination suit.If she wins the case,she will make $40,000;but if she loses,she gets nothing.Assume that she has to spend $5000 preparing the case.What is her expected gain?
A)$35,000
B)$7,000
C)$10,500
D)$9,000

Accepted Answer

The expected gain is calculated as (0.3 * $40,000) - $5,000 = $12,000 - $5,000 = $7,000.

Question 13

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the probability that at least 3 of the individuals have group A blood.
A).8208
B).5443
C).4557
D).1792

Accepted Answer

To find the probability of at least 3 individuals having group A blood, we need to find the probability of 3, 4, 5, and 6 individuals having group A blood and add them up. Using the binomial probability formula, we get:

P(3) = (6 choose 3)(0.4^3)(0.6^3) = 0.2304
P(4) = (6 choose 4)(0.4^4)(0.6^2) = 0.2508
P(5) = (6 choose 5)(0.4^5)(0.6^1) = 0.1536
P(6) = (6 choose 6)(0.4^6)(0.6^0) = 0.0256

Adding them up, we get:

P(at least 3) = P(3) + P(4) + P(5) + P(6) = 0.6604

Multiplying by 100% to convert to percentage, we get:

P(at least 3) = 66.04%

Rounding to the nearest tenth, we get:

P(at least 3) = 66.0%

Therefore, the best choice is C (66.0%).

Question 14

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day.Calculate the standard deviation of the number delivered when 8 local letters are mailed.
A).85
B).72
C)2.68
D)2.83

Accepted Answer

σ ==== .85

Question 15

According to data from the state blood program,40 percent of all individuals have group A blood.Suppose that of six randomly selected individuals,three have group A blood.Would you believe the data from the state blood program?
A)Yes,probability is > .05
B)Yes,probability is < .05
C)No

Accepted Answer

Yes, the probability is greater than 0.05, and therefore we do not reject the null hypothesis.

Question 16

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?
A)1.5
B)1.22
C)0.75
D)0.87

Accepted Answer

The possible outcomes of tossing a fair coin three times are: HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.
Since the coin is fair, each outcome has probability of 1/8. The expected value of X is E(X) = 3/2. 
To calculate the variance, we use the formula Var(X) = E(X^2) - [E(X)]^2.
We can find E(X^2) as follows:
E(X^2) = (0^2 + 1^2 + 2^2 + 3^2)/8 = 14/8 = 1.75
Therefore, Var(X) = 1.75 - (3/2)^2 = 1.75 - 2.25 = 0.75. 
Thus, the answer is C.

Question 17

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,find the mean number of troubles repaired on the same day.
A)3.75
B)0.94
C)1.94
D)2.50

Accepted Answer

The mean number of troubles repaired on the same day can be found using the formula for the mean of a binomial distribution: mean = n * p, where n is the number of trials (5) and p is the probability of success (0.75). Thus, mean = 5 * 0.75 = 3.75.

Question 18

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.
A)1.5
B)1.0
C)2.0
D)1.1

Accepted Answer

E[X] == 0+ (1)+ (2)+ (3)= 1.5

Question 19

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day.If you mail eight local letters,what is the average number you expect to be delivered the next day?
A)3.6
B)4.0
C)7.2
D)2.7

Accepted Answer

μ_x = np = (8)(.9) = 7.2

Question 20

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the probability that exactly three of the individuals have group A blood.
A).4000
B).2765
C).5875
D).0041

Accepted Answer

The probability of an individual having group A blood is 0.4. The probability of an individual not having group A blood is 0.6. The probability of exactly three individuals having group A blood is given by the binomial distribution:P(X = 3) = (6 choose 3) * (0.4)^3 * (0.6)^3 = 0.2765Therefore, the correct answer is B.

Quiz 5: Discrete Random Variables

Your company's internal auditor believes that 10 percent of the company's invoices contain errors.To check this theory,20 invoices are randomly selected,and 5 are found to have errors.What is the probability that of the 20 invoices selected,5 or more would contain errors if the theory is valid?

The Post Office has established a record in a major midwestern city for delivering 90 percent 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?

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the probability that none of the individuals has group A blood.

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?

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day.When there are 8 local letters mailed,what is the probability that the number delivered will be within 2 standard deviations of the mean?

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the mean number of individuals having group A blood.

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 standard deviation for this distribution?

A lawyer believes that the probability is .3 that she can win a discrimination suit.If she wins the case,she will make $40,000;but if she loses,she gets nothing.Assume that she has to spend $5000 preparing the case.What is her expected gain?

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the probability that at least 3 of the individuals have group A blood.

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day.Calculate the standard deviation of the number delivered when 8 local letters are mailed.

According to data from the state blood program,40 percent of all individuals have group A blood.Suppose that of six randomly selected individuals,three have group A blood.Would you believe the data from the state blood program?

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?

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.

The Post Office has established a record in a major midwestern city for delivering 90 percent of its local mail the next working day.If you mail eight local letters,what is the average number you expect to be delivered the next day?

According to data from the state blood program,40 percent of all individuals have group A blood.If six individuals give blood,find the probability that exactly three of the individuals have group A blood.