Question 1

A computer salesman averages 1.9 sales per week.Use the Poisson distribution to find the probability that in a randomly selected week the number of computers sold is 1.

Accepted Answer

A) 0.2842 
B) 0.3552 
C) 0.7158 
D) 0.3126 
E) 0.5399 
A) 0.2842 
B) 0.3552 
C) 0.7158 
D) 0.3126 
E) 0.5399

Question 2

In one city,police estimate that 78% of drivers wear their seat belts.They set up a safety roadblock and they stop drivers to check for seat belt use.If 160 drivers are stopped,what is the probability they find at least 30 not wearing their seat belts?

Accepted Answer

A) 0.575 
B) 0.161 
C) 0.788 
D) 0.839 
E) 0.425 
A) 0.575 
B) 0.161 
C) 0.788 
D) 0.839 
E) 0.425

Question 3

The number of emergency calls received by a switchboard in an hour has an expected value of 0.5 and a standard deviation of 0.7.Find the mean and standard deviation of the number of emergency calls received by the switch board during a workshift lasting 4 hours.Assume that the hours are independent of each other.

Accepted Answer

A) μ = 2,σ = 11.2 
B) μ = 1.00,σ = 2.8 
C) μ = 2,σ = 2.8 
D) μ = 2,σ = 1.40 
E) μ = 1.00,σ = 1.40 
A) μ = 2,σ = 11.2 
B) μ = 1.00,σ = 2.8 
C) μ = 2,σ = 2.8 
D) μ = 2,σ = 1.40 
E) μ = 1.00,σ = 1.40

Question 4

Suppose the probability of contracting a certain disease is p = 0.0006 for a new case in a given year.Use the Poisson model to approximate the probability that in a town where 8000 uninfected people there will be at least one new case of the disease next year.

Accepted Answer

A) 0.9523 
B) 0.9918 
C) 0.0395 
D) 0.0082 
E) 0.9994 
A) 0.9523 
B) 0.9918 
C) 0.0395 
D) 0.0082 
E) 0.9994

Question 5

Suppose that in one town adult men have a mean weight of 77 kg with a standard deviation of 7 kg.Adult women have a mean weight of 63 kg with a standard deviation of 5 kg.10 year old children have a mean weight of 41 kg with a standard deviation of 3 kg.Suppose that a man,a woman,and a 10-year old child get into an elevator.What are the mean and standard deviation of their total weight?

Accepted Answer

A) μ = 107 kg,σ = 9.11 kg 
B) μ = 181 kg,σ = 15 kg 
C) μ = 181 kg,σ = 9.11 kg 
D) μ = 107 kg,σ = 15 kg 
E) μ = 181 kg,σ = 7.33 kg 
A) μ = 107 kg,σ = 9.11 kg 
B) μ = 181 kg,σ = 15 kg 
C) μ = 181 kg,σ = 9.11 kg 
D) μ = 107 kg,σ = 15 kg 
E) μ = 181 kg,σ = 7.33 kg

Question 6

In one town,the number of burglaries in a week has a Poisson distribution with a mean of 2.1.Find the probability that in a randomly selected week the number of burglaries is at least three.

Accepted Answer

A) 0.1614 
B) 0.1890 
C) 0.6496 
D) 0.8110 
E) 0.3504 
A) 0.1614 
B) 0.1890 
C) 0.6496 
D) 0.8110 
E) 0.3504

Question 7

An insurance policy costs $150 per year,and will pay policyholders $16,000 if they suffer a major injury (resulting in hospitalization)or $6000 if they suffer a minor injury (resulting in lost time from work).The company estimates that each year 1 in every 2000 policyholders will have a major injury and 1 in every 400 a minor injury.What is the standard deviation of the company's profit on this policy?

Accepted Answer

A) $582.92 
B) $466.34 
C) $447.68 
D) $498.98 
E) $550.28 
A) $582.92 
B) $466.34 
C) $447.68 
D) $498.98 
E) $550.28

Question 8

An insurance company estimates that it should make an annual profit of $140 on each homeowner's policy written,with a standard deviation of $5500.If it writes 10,000 of these policies,what are the mean and standard deviation of the annual profit? Assume that policies are independent of each other.

Accepted Answer

A) μ = $14,000.00,σ = $550,000.00 
B) μ = $1,400,000,σ = $550,000,000,000 
C) μ = $1,400,000,σ = $550,000.00 
D) μ = $1,400,000,σ = $55,000,000 
E) μ = $14,000.00,σ = $55,000,000 
A) μ = $14,000.00,σ = $550,000.00 
B) μ = $1,400,000,σ = $550,000,000,000 
C) μ = $1,400,000,σ = $550,000.00 
D) μ = $1,400,000,σ = $55,000,000 
E) μ = $14,000.00,σ = $55,000,000

Question 9

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $$X _ { 1 } + X _ { 2 }$$ $$\begin{array} { c | c c } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 14 & 3 \
\mathrm { Y } & 60 & 8
\end{array}$$

Accepted Answer

A) ? = 28,? = 4.24 
B) ? = 74,? = 8.54 
C) ? = 28,? = 6 
D) ? = 28,? = 2.45 
E) ? = 74,? = 11 
A) ? = 28,? = 4.24 
B) ? = 74,? = 8.54 
C) ? = 28,? = 6 
D) ? = 28,? = 2.45 
E) ? = 74,? = 11

Question 10

A beginning archer is able to hit the bull's-eye 41% of the time.If she shoots 7 arrows,what is the probability that she gets at most 3 bull's-eyes? Assume each shot is independent of the others.

Accepted Answer

A) 0.6906 
B) 0.0625 
C) 0.3094 
D) 0.2923 
E) 0.3983 
A) 0.6906 
B) 0.0625 
C) 0.3094 
D) 0.2923 
E) 0.3983

Question 11

The number of calls received by a mountain search and rescue team in a day has a Poisson distribution with a mean of 0.71.Find the probability that on a randomly selected day,they will receive fewer than two calls.

Accepted Answer

A) 0.8761 
B) 0.3491 
C) 0.8407 
D) 0.1239 
E) 0.1593 
A) 0.8761 
B) 0.3491 
C) 0.8407 
D) 0.1239 
E) 0.1593

Question 12

A study found that 56% of people working for large companies are dissatisfied with their job.Your manager conducts a survey at your company and finds that 225 of the 440 employees are dissatisfied with their job.

Accepted Answer

A) Yes. 
B) No.Employees within the same company are not independent of one another. 
C) No.There are more than two possible outcomes. 
D) No.The chance of getting a dissatisfied employee changes depending on who has already been selected. 
E) No.225 is more than 10% of 440. 
A) Yes. 
B) No.Employees within the same company are not independent of one another. 
C) No.There are more than two possible outcomes. 
D) No.The chance of getting a dissatisfied employee changes depending on who has already been selected. 
E) No.225 is more than 10% of 440.

Question 13

The probability that a radish seed will germinate is 0.7.A gardener plants seeds in batches of 4.Find the standard deviation of the number of seeds germinating in each batch.

Accepted Answer

A) 0.794 
B) 0.784 
C) 0.210 
D) 0.906 
E) 0.917 
A) 0.794 
B) 0.784 
C) 0.210 
D) 0.906 
E) 0.917

Question 14

A basketball player usually makes 58% of his free shots.Tonight he made 10 shots in a row.Is this evidence that he is on a winning streak? That is,is this streak so unusual that it means the probability he makes a shot must have changed? Explain.

Accepted Answer

A) No; if his success rate were still 58%,he would have a 5.8% chance of making 10 shots in a row.That's not a highly unusual result. 
B) Yes; if his success rate were still 58%,he would have only a 5.8% chance of making 10 shots in a row.That's an unusual result. 
C) Yes; if his success rate were still 58%,he would have only a 0.2% chance of making 10 shots in a row.That's an unusual result. 
D) Yes; if his success rate were still 58%,he would have only a 0.4% chance of making 10 shots in a row.That's an unusual result. 
E) No; if his success rate were still 58%,he would have a 0.4% chance of making 10 shots in a row.That's not an unusual result. 
A) No; if his success rate were still 58%,he would have a 5.8% chance of making 10 shots in a row.That's not a highly unusual result. 
B) Yes; if his success rate were still 58%,he would have only a 5.8% chance of making 10 shots in a row.That's an unusual result. 
C) Yes; if his success rate were still 58%,he would have only a 0.2% chance of making 10 shots in a row.That's an unusual result. 
D) Yes; if his success rate were still 58%,he would have only a 0.4% chance of making 10 shots in a row.That's an unusual result. 
E) No; if his success rate were still 58%,he would have a 0.4% chance of making 10 shots in a row.That's not an unusual result.

Question 15

Suppose a computer chip manufacturer rejects 4% of the chips produced because they fail presale testing.If 135 chips are picked at random,is it appropriate to use a Normal model to approximate the distribution of the number of bad chips?

Accepted Answer

A) No; Normal model cannot be used to approximate the distribution because np < 10. 
B) Yes; Normal model with μ = 129.6 and σ = 2.28 can be used to approximate the distribution. 
C) Yes; Normal model with μ = 5.4 and σ = 2.28 can be used to approximate the distribution. 
D) No; Normal model cannot be used to approximate the distribution because nq < 10. 
E) Yes; Normal model with μ = 5.4 and σ = 5.18 can be used to approximate the distribution. 
A) No; Normal model cannot be used to approximate the distribution because np < 10. 
B) Yes; Normal model with μ = 129.6 and σ = 2.28 can be used to approximate the distribution. 
C) Yes; Normal model with μ = 5.4 and σ = 2.28 can be used to approximate the distribution. 
D) No; Normal model cannot be used to approximate the distribution because nq < 10. 
E) Yes; Normal model with μ = 5.4 and σ = 5.18 can be used to approximate the distribution.

Question 16

The probability model below describes the number of thunderstorms that a certain town may experience during the month of August. $$\begin{array} { l | c c c c } 
\text { Number of storms } & 0 & 1 & 2 & 3 \
\hline \text { Probability } & 0.1 & 0.2 & 0.4 & 0.3
\end{array}$$ What is the standard deviation of the number of storms in August?

Accepted Answer

A) 0.88 
B) 0.94 
C) 0.66 
D) 0.75 
E) 0.77 
A) 0.88 
B) 0.94 
C) 0.66 
D) 0.75 
E) 0.77

Question 17

A company bids on two contracts.It anticipates a profit of $70,000 if it gets the larger contract and a profit of $30,000 if it gets the smaller contract.It estimates that there's a 10% chance of winning the larger contract and a 60% chance of winning the smaller contract.Find the standard deviation of the company's profit.Assume that the contracts will be awarded independently.

Accepted Answer

A) $30,758 
B) $24,863 
C) $25,632 
D) $29,477 
E) $28,195 
A) $30,758 
B) $24,863 
C) $25,632 
D) $29,477 
E) $28,195

Question 18

A contractor is considering a sale that promises a profit of $24,000 with a probability of 0.7 or a loss (due to bad weather,strikes,and such)of $16,000 with a probability of 0.3.What is the expected profit?

Accepted Answer

A) $8000 
B) $16,800 
C) $28,000 
D) $21,600 
E) $12,000 
A) $8000 
B) $16,800 
C) $28,000 
D) $21,600 
E) $12,000

Question 19

Suppose that in one town,50-year old men have a mean weight of 80 kg with a standard deviation of 7 kg.30-year old men have a mean weight of 71 kg with a standard deviation of 6 kg.How much heavier do you expect a 50-year old man to be than a 30-year old man and what is the standard deviation of this difference?

Accepted Answer

A) 80 kg,7 kg 
B) 9 kg,3.61 kg 
C) 151 kg,4.31 kg 
D) 9 kg,9.22 kg 
E) 9 kg,13 kg 
A) 80 kg,7 kg 
B) 9 kg,3.61 kg 
C) 151 kg,4.31 kg 
D) 9 kg,9.22 kg 
E) 9 kg,13 kg

Question 20

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X + 2Y.Round to two decimal places if necessary. $$\begin{array} { l | l l } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 60 & 12 \
\mathrm { Y } & 50 & 7
\end{array}$$

Accepted Answer

A) ? = 110,? = 18.44 
B) ? = 100,? = 18.44 
C) ? = 110,? = 26 
D) ? = 160,? = 18.44 
E) ? = 160,? = 26 
A) ? = 110,? = 18.44 
B) ? = 100,? = 18.44 
C) ? = 110,? = 26 
D) ? = 160,? = 18.44 
E) ? = 160,? = 26

Exam 14: Random Variables

A computer salesman averages 1.9 sales per week.Use the Poisson distribution to find the probability that in a randomly selected week the number of computers sold is 1.

In one city,police estimate that 78% of drivers wear their seat belts.They set up a safety roadblock and they stop drivers to check for seat belt use.If 160 drivers are stopped,what is the probability they find at least 30 not wearing their seat belts?

Suppose the probability of contracting a certain disease is p = 0.0006 for a new case in a given year.Use the Poisson model to approximate the probability that in a town where 8000 uninfected people there will be at least one new case of the disease next year.

In one town,the number of burglaries in a week has a Poisson distribution with a mean of 2.1.Find the probability that in a randomly selected week the number of burglaries is at least three.

An insurance company estimates that it should make an annual profit of $140 on each homeowner's policy written,with a standard deviation of $5500.If it writes 10,000 of these policies,what are the mean and standard deviation of the annual profit? Assume that policies are independent of each other.

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X1+X2X _ { 1 } + X _ { 2 }X1​+X2​ Mean SD 14 3 60 8

A beginning archer is able to hit the bull's-eye 41% of the time.If she shoots 7 arrows,what is the probability that she gets at most 3 bull's-eyes? Assume each shot is independent of the others.

The number of calls received by a mountain search and rescue team in a day has a Poisson distribution with a mean of 0.71.Find the probability that on a randomly selected day,they will receive fewer than two calls.

A study found that 56% of people working for large companies are dissatisfied with their job.Your manager conducts a survey at your company and finds that 225 of the 440 employees are dissatisfied with their job.

The probability that a radish seed will germinate is 0.7.A gardener plants seeds in batches of 4.Find the standard deviation of the number of seeds germinating in each batch.

A basketball player usually makes 58% of his free shots.Tonight he made 10 shots in a row.Is this evidence that he is on a winning streak? That is,is this streak so unusual that it means the probability he makes a shot must have changed? Explain.

Suppose a computer chip manufacturer rejects 4% of the chips produced because they fail presale testing.If 135 chips are picked at random,is it appropriate to use a Normal model to approximate the distribution of the number of bad chips?

The probability model below describes the number of thunderstorms that a certain town may experience during the month of August. Number of storms 0 1 2 3 Probability 0.1 0.2 0.4 0.3 What is the standard deviation of the number of storms in August?

A contractor is considering a sale that promises a profit of $24,000 with a probability of 0.7 or a loss (due to bad weather,strikes,and such)of $16,000 with a probability of 0.3.What is the expected profit?

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable X + 2Y.Round to two decimal places if necessary. Mean SD 60 12 50 7

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Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable $X _ { 1 } + X _ { 2 }$ Mean SD 14 3 60 8