Question 1

$$\begin{array} { r | r } 
\mathrm { x } & \mathrm { P } ( \mathrm { X } = \mathrm { x } ) \
\hline 0 & 0.50 \
1 & 0.12 \
2 & 0.16 \
3 & 0.20 \
4 & 0.02
\end{array}$$

Accepted Answer

A) 1.12 
B) 0.97 
C) 1.52 
D) 1.02 
E) 1.62 
A) 1.12 
B) 0.97 
C) 1.52 
D) 1.02 
E) 1.62

Question 2

$$\begin{array} { c | c c c r } 
\mathrm { x } & 3 & 6 & 9 & 12 \
\hline \mathrm { P } ( \mathrm { X } = \mathrm { x } ) & 0.1 & 0.4 & 0.2 & 0.3
\end{array}$$

Accepted Answer

A) 8.62 
B) 2.71 
C) 2.41 
D) 3.01 
E) 3.32 
A) 8.62 
B) 2.71 
C) 2.41 
D) 3.01 
E) 3.32

Question 3

A company manufactures batteries in batches of 23 and there is a 3% rate of defects.Find the standard deviation of the number of defects per batch.

Accepted Answer

A) 0.800 
B) 0.291 
C) 0.815 
D) 0.831 
E) 0.818 
A) 0.800 
B) 0.291 
C) 0.815 
D) 0.831 
E) 0.818

Question 4

The number of golf balls ordered by customers of a pro shop has the following probability distribution. $$\begin{array} { r | r | r | r | r | r } 
\mathrm { x } & 3 & 6 & 9 & 12 & 15 \
\hline \mathrm { p } ( \mathrm { x } ) & 0.14 & 0.12 & 0.36 & 0.28 & 0.10
\end{array}$$

Accepted Answer

A) 6.57 
B) 8.28 
C) 9 
D) 9.24 
E) 8.82 
A) 6.57 
B) 8.28 
C) 9 
D) 9.24 
E) 8.82

Question 5

A slot machine at a casino pays out an average of $0.93,with a standard deviation of $120.It costs a dollar to play.If a person plays 10 times,what are the mean and standard deviation of the casino's profit?

Accepted Answer

A) μ = $9.30,σ = $379.47 
B) μ = $0.70,σ = $12,000 
C) μ = $9.30,σ = $1200 
D) μ = $0.70,σ = $1200 
E) μ = $0.70,σ = $379.47 
A) μ = $9.30,σ = $379.47 
B) μ = $0.70,σ = $12,000 
C) μ = $9.30,σ = $1200 
D) μ = $0.70,σ = $1200 
E) μ = $0.70,σ = $379.47

Question 6

Suppose a computer chip manufacturer rejects 5% of the chips produced because they fail presale testing.If we select 50 chips at random,how many chips do you expect to fail the testing?

Accepted Answer

A) 0.05 chips 
B) 21.8 chips 
C) 2.5 chips 
D) 5 chips 
E) 2 chips 
A) 0.05 chips 
B) 21.8 chips 
C) 2.5 chips 
D) 5 chips 
E) 2 chips

Question 7

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 4Y - 5.Round to two decimal places if necessary. $$\begin{array} { l | l l } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 190 & 16 \
\mathrm { Y } & 280 & 14
\end{array}$$

Accepted Answer

A) ? = 1115,? = 55.78 
B) ? = 1120,? = 56.22 
C) ? = 1115,? = 51 
D) ? = 1120,? = 56 
E) ? = 1115,? = 56 
A) ? = 1115,? = 55.78 
B) ? = 1120,? = 56.22 
C) ? = 1115,? = 51 
D) ? = 1120,? = 56 
E) ? = 1115,? = 56

Question 8

The probability model below describes the number of thunderstorms that a certain town may experience during the month of August. $$\begin{array} { l | l l l l } 
\text { Number of storms } & 0 & 1 & 2 & 3 \
\hline \text { Probability } & 0.1 & 0.3 & 0.5 & 0.1
\end{array}$$ How many storms can the town expect each August?

Accepted Answer

A) 1.5 
B) 1.9 
C) 2.0 
D) 1.7 
E) 1.6 
A) 1.5 
B) 1.9 
C) 2.0 
D) 1.7 
E) 1.6

Question 9

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $30 plus an extra $60 for the ace of hearts.For any other card you win nothing.Find the expected amount you will win.

Accepted Answer

A) $8.08 
B) $4.62 
C) $6.92 
D) $6.35 
E) $7.50 
A) $8.08 
B) $4.62 
C) $6.92 
D) $6.35 
E) $7.50

Question 10

A company is interviewing applicants for managerial positions.They plan to hire two people.They have already rejected most candidates and are left with a group of 9 applicants of whom 4 are women.Unable to differentiate further between the applicants,they choose two people at random from this group of 9.Let the random variable X be the number of men that are chosen.Find the probability model for X.

Accepted Answer

A)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.167 & 0.410 & 0.278
\end{array}$$ 
B)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.198 & 0.494 & 0.309
\end{array}$$ 
C)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.278 & 0.556 & 0.167
\end{array}$$ 
D)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.167 & 0.278 & 0.278
\end{array}$$ 
E)  $$\begin{array} { c | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 11.077 & 66.462 & 66.462
\end{array}$$ 
A)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.167 & 0.410 & 0.278
\end{array}$$ 
B)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.198 & 0.494 & 0.309
\end{array}$$ 
C)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.278 & 0.556 & 0.167
\end{array}$$ 
D)  $$\begin{array} { l | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 0.167 & 0.278 & 0.278
\end{array}$$ 
E)  $$\begin{array} { c | c c c } 
\text { Number men } & 0 & 1 & 2 \
\hline \text { P(Number men) } & 11.077 & 66.462 & 66.462
\end{array}$$

Question 11

Miguel buys a large bottle and a small bottle of juice.The amount of juice that the manufacturer puts in the large bottle is a random variable with a mean of 1024 ml and a standard deviation of 12 ml.The amount of juice that the manufacturer puts in the small bottle is a random variable with a mean of 504 ml and a standard deviation of 2 ml.If the total amount of juice in the two bottles can be described by a Normal model,what is the probability that the total amount of juice in the two bottles is more than 1554.8 ml?

Accepted Answer

A) 0.014 
B) 0.011 
C) 0.018 
D) 0.986 
E) 0.982 
A) 0.014 
B) 0.011 
C) 0.018 
D) 0.986 
E) 0.982

Question 12

In the 4 × 100 relay event,each of four runners runs 100 metres.A college team is preparing for a competition.The means and standard deviations of the times (in seconds)of their four runners are shown in the table: $$\begin{array} { c | c | c } 
\text { Runner } & \text { Mean } & \text { SD } \
\hline 1 & 12.01 & 0.10 \
2 & 12.26 & 0.15 \
3 & 11.93 & 0.10 \
4 & 11.81 & 0.08
\end{array}$$ Find the probability that in the competition they will beat their previous best time of 47.501 sec.Assume that the runners' performances are independent and that each runner's time follows a Normal model.

Accepted Answer

A) 0.036 
B) 0.982 
C) 0.018 
D) 0.989 
E) 0.011 
A) 0.036 
B) 0.982 
C) 0.018 
D) 0.989 
E) 0.011

Question 13

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable Y - 19.Round to two decimal places if necessary. $$\begin{array} { l | l l } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 300 & 29 \
\mathrm { Y } & 220 & 30
\end{array}$$

Accepted Answer

A) ? = 201,? = 35.51 
B) ? = 201,? = 11 
C) ? = 201,? = 23.22 
D) ? = 220,? = 30 
E) ? = 201,? = 30 
A) ? = 201,? = 35.51 
B) ? = 201,? = 11 
C) ? = 201,? = 23.22 
D) ? = 220,? = 30 
E) ? = 201,? = 30

Question 14

An insurance company estimates that it should make an annual profit of $150 on each homeowner's policy written,with a standard deviation of $6000.If it writes three of these policies,the mean annual profit for the company is found to be $3 \times 150 = 450$ The standard deviation of the company's annual profit is found to be $\sqrt { 6000 ^ { 2 } + 6000 ^ { 2 } + 6000 ^ { 2 } } \approx \$ 10,392 .$ What assumptions (if any)underlie the calculation of the mean? of the standard deviation?

Accepted Answer

A) Mean: that the profit on each policy follows a Normal model Standard deviation: that the policies are independent of one another and that the profit on each policy follows a Normal model 
B) Mean: no assumptions required Standard deviation: no assumptions required 
C) Mean: no assumptions required Standard deviation: that the policies are independent of one another and that the profit on each policy follows a Normal model 
D) Mean: no assumptions required Standard deviation: that the policies are independent of one another 
E) Mean: that the policies are independent of one another Standard deviation: that the policies are independent of one another 
A) Mean: that the profit on each policy follows a Normal model Standard deviation: that the policies are independent of one another and that the profit on each policy follows a Normal model 
B) Mean: no assumptions required Standard deviation: no assumptions required 
C) Mean: no assumptions required Standard deviation: that the policies are independent of one another and that the profit on each policy follows a Normal model 
D) Mean: no assumptions required Standard deviation: that the policies are independent of one another 
E) Mean: that the policies are independent of one another Standard deviation: that the policies are independent of one another

Question 15

A pool of possible jurors consists of 15 men and 18 women.A jury of 12 is picked at random from this group.What is the probability that the jury contains all women?

Accepted Answer

A) Yes 
B) No.15 is more than 10% of 18 
C) Yes,assuming the possible jurors are unrelated. 
D) No.There are more than two possible outcomes. 
E) No.The chance of a woman changes depending on who has already been picked. 
A) Yes 
B) No.15 is more than 10% of 18 
C) Yes,assuming the possible jurors are unrelated. 
D) No.There are more than two possible outcomes. 
E) No.The chance of a woman changes depending on who has already been picked.

Question 16

A company sells light bulbs in packages of 20 and estimates that the mean number of defective light bulbs in a package is 0.6 with a standard deviation of 0.7.If a customer buys 12 packages,what are the expected value and standard deviation of the number of defective light bulbs? Assume that packages are independent of each other.

Accepted Answer

A) μ = 7.2,σ = 2.42 
B) μ = 7.2,σ = 100.8 
C) μ = 2.08,σ = 8.4 
D) μ = 7.2,σ = 8.4 
E) μ = 2.08,σ = 2.42 
A) μ = 7.2,σ = 2.42 
B) μ = 7.2,σ = 100.8 
C) μ = 2.08,σ = 8.4 
D) μ = 7.2,σ = 8.4 
E) μ = 2.08,σ = 2.42

Question 17

The probability that a call received by a certain switchboard will be a wrong number is 0.02.Use the Poisson approximation to the binomial distribution to find the probability that among 140 calls received by the switchboard,there are at least two wrong numbers.

Accepted Answer

A) 0.5305 
B) 0.7616 
C) 0.7689 
D) 0.2384 
E) 0.2311 
A) 0.5305 
B) 0.7616 
C) 0.7689 
D) 0.2384 
E) 0.2311

Question 18

A company is interviewing applicants for managerial positions.They plan to hire two people.They have already rejected most candidates and are left with a group of 10 applicants of whom 4 are women.Unable to differentiate further between the applicants,they choose two people at random from this group of 10.Let the random variable X be the number of men that are chosen.Find the expected value of X.

Accepted Answer

A) μ = 1.50 
B) μ = 1.20 
C) μ = 0.80 
D) μ = 0.57 
E) μ = 0.93 
A) μ = 1.50 
B) μ = 1.20 
C) μ = 0.80 
D) μ = 0.57 
E) μ = 0.93

Question 19

We wish to predict the outcome of a local mayoral election,and poll 500 of the 4000 likely voters to see how many favor the incumbent.

Accepted Answer

A) Yes,assuming the 500 people are unrelated. 
B) No.The probability of success is the same on every trial. 
C) No.The responses are independent. 
D) No.There are more than two possible outcomes. 
E) No.500 is more than 10% of 4000. 
A) Yes,assuming the 500 people are unrelated. 
B) No.The probability of success is the same on every trial. 
C) No.The responses are independent. 
D) No.There are more than two possible outcomes. 
E) No.500 is more than 10% of 4000.

Question 20

Hugh buys $8000 worth of stock in an electronics company which he hopes to sell afterward at a profit.The company is developing a new laptop computer and a new desktop computer.If it releases both computers before its competitor,the value of Hugh's stock will jump to $21,000.If it releases one of the computers before its competitor,the value of Hugh's stock will jump to $17,000.If it fails to release either computer before its competitor,Hugh's stock will be worth only $5000.Hugh believes that there is a 40% chance that the company will release the laptop before its competitor and a 50% chance that the company will release the desktop before its competitor. Create a probability model for Hugh's profit.Assume that the development of the laptop and the development of the desktop are independent events.

Accepted Answer

A)  $$\begin{array} { l | c c c } 
\text { Profit } & \$ 21,000 & \$ 17,000 & \$ 5000 \
\hline \text { P(Profit } ) & 0.2 & 0.5 & 0.3
\end{array}$$ 
B)  $$\begin{array} { | l | c c c | } 
\hline \text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \
\hline \text { P(Profit } ) & 0.2 & 0.5 & 0.3
\end{array}$$ 
C)  $$\begin{array} { | l | c c c | } 
\hline \text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \
\hline \text { P(Profit } ) & 0.2 & 0.2 & 0.3
\end{array}$$ 
D)  $$\begin{array} { l | c c c } 
\text { Profit } & \$ 21,000 & \$ 17,000 & \$ 5000 \
\hline \text { P(Profit) } & 0.2 & 0.9 & 0.3
\end{array}$$ 
E)  $$\begin{array} {            l | c c c } 
\text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \
\hline \text { P(Profit } ) & 0.9 & 0.2 & 0.3
\end{array}$$ 
A)  $$\begin{array} { l | c c c } 
\text { Profit } & \$ 21,000 & \$ 17,000 & \$ 5000 \
\hline \text { P(Profit } ) & 0.2 & 0.5 & 0.3
\end{array}$$ 
B)  $$\begin{array} { | l | c c c | } 
\hline \text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \
\hline \text { P(Profit } ) & 0.2 & 0.5 & 0.3
\end{array}$$ 
C)  $$\begin{array} { | l | c c c | } 
\hline \text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \
\hline \text { P(Profit } ) & 0.2 & 0.2 & 0.3
\end{array}$$ 
D)  $$\begin{array} { l | c c c } 
\text { Profit } & \$ 21,000 & \$ 17,000 & \$ 5000 \
\hline \text { P(Profit) } & 0.2 & 0.9 & 0.3
\end{array}$$ 
E)  $$\begin{array} {            l | c c c } 
\text { Profit } & \$ 13,000 & \$ 9000 & - \$ 3000 \
\hline \text { P(Profit } ) & 0.9 & 0.2 & 0.3
\end{array}$$

Exam 14: Random Variables

(=) 0 0.50 1 0.12 2 0.16 3 0.20 4 0.02

3 6 9 12 (=) 0.1 0.4 0.2 0.3

A company manufactures batteries in batches of 23 and there is a 3% rate of defects.Find the standard deviation of the number of defects per batch.

The number of golf balls ordered by customers of a pro shop has the following probability distribution. 3 6 9 12 15 () 0.14 0.12 0.36 0.28 0.10

A slot machine at a casino pays out an average of $0.93,with a standard deviation of $120.It costs a dollar to play.If a person plays 10 times,what are the mean and standard deviation of the casino's profit?

Suppose a computer chip manufacturer rejects 5% of the chips produced because they fail presale testing.If we select 50 chips at random,how many chips do you expect to fail the testing?

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 4Y - 5.Round to two decimal places if necessary. Mean SD 190 16 280 14

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 0.3 0.5 0.1 How many storms can the town expect each August?

You pick a card from a deck.If you get a face card,you win $15.If you get an ace,you win $30 plus an extra $60 for the ace of hearts.For any other card you win nothing.Find the expected amount you will win.

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable Y - 19.Round to two decimal places if necessary. Mean SD 300 29 220 30

A pool of possible jurors consists of 15 men and 18 women.A jury of 12 is picked at random from this group.What is the probability that the jury contains all women?

The probability that a call received by a certain switchboard will be a wrong number is 0.02.Use the Poisson approximation to the binomial distribution to find the probability that among 140 calls received by the switchboard,there are at least two wrong numbers.

We wish to predict the outcome of a local mayoral election,and poll 500 of the 4000 likely voters to see how many favor the incumbent.

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