Question 1

On one tropical island,hurricanes occur with a mean of 2.37 per year.Assuming that the number of hurricanes can be modeled by a Poisson distribution,find the probability that during the next 2 years the number of hurricanes will be 3.

Accepted Answer

A) 0.2074 
B) 0.0935 
C) 0.1838 
D) 0.1551 
E) 0.0194

Question 2

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 $\sqrt { 6000 ^ { 2 } + 6000 ^ { 2 } + 6000 ^ { 2 } } \approx \$ 10,392 .$ The calculation of the standard deviation requires the assumption that the policies are independent of one another.Which of the following examples describes a situation in which that assumption may be violated? A: The houses are close to each other and may all be affected by the same catastrophe such as earthquake,flooding,or fire.
B: The owners all bought their homes in the same year and may all be affected by a downturn in the market.
C: The owners of the three homes are related.
D: All three houses were bought for the same price.

Accepted Answer

A) B only 
B) A,B,C,and D 
C) A and C 
D) A only 
E) B and D

Question 3

Find the standard deviation for the given probability distribution. $$\begin{array} { c | c } 
\mathrm { x } & \mathrm { P } ( \mathrm { x } ) \
\hline 0 & 0.27 \
\hline 1 & 0.07 \
\hline 2 & 0.07 \
\hline 3 & 0.22 \
\hline 4 & 0.37
\end{array}$$

Accepted Answer

A) 2.87 
B) 1.28 
C) 1.73 
D) 2.73 
E) 1.65

Question 4

You have arranged to go camping for two days in March.You believe that the probability that it will rain on the first day is 0.4.If it rains on the first day,the probability that it also rains on the second day is 0.5.If it doesn't rain on the first day,the probability that it rains on the second day is 0.3.Let the random variable X be the number of rainy days during your camping trip.Find the standard deviation of X.

Accepted Answer

A) σ = 0.67 
B) σ = 0.76 
C) σ = 0.57 
D) σ = 0.77 
E) σ = 0.70

Question 5

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 4X + 19.Round to two decimal places if necessary. $$\begin{array} { l | l l } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 100 & 14 \
\mathrm { Y } & 120 & 7
\end{array}$$

Accepted Answer

A) ? = 400,? = 75 
B) ? = 419,? = 75 
C) ? = 400,? = 56 
D) ? = 419,? = 59.14 
E) ? = 419,? = 56

Question 6

A tennis player makes a successful first serve 70% of the time.If she serves 80 times,is it appropriate to use a Normal model to approximate the distribution of the number of good first serves? Assume that each serve is independent of the others.

Accepted Answer

A) No; Normal model cannot be used to approximate the distribution because nq < 10. 
B) Yes; Normal model with μ = 56 and σ = 16.80 can be used to approximate the distribution. 
C) Yes; Normal model with μ = 56 and σ = 4.10 can be used to approximate the distribution. 
D) Yes; Normal model with μ = 24 and σ = 16.80 can be used to approximate the distribution. 
E) No; Normal model cannot be used to approximate the distribution because np < 10.

Question 7

A carnival game offers a $120 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $9 to play and you're willing to spend up to $36 trying to win.You estimate that you have a 10% chance of hitting the balloon on any throw.Find the expected amount you will win.Assume that throws are independent of each other.

Accepted Answer

A) $13.01 
B) $10.32 
C) -$14.76 
D) -$11.16 
E) $19.32

Question 8

Suppose the probability of contracting a certain disease is p = 0.0004 for a new case in a given year.What is the expected number of new cases in a year in a town where 4000 people do not currently have the disease?

Accepted Answer

A) 0.16 
B) 1.6 
C) 16 
D) 0.0004 
E) 0.10

Question 9

In the 4 × 100 relay event,each of four runners runs 100 metres.A university team is preparing for a competition.The means and standard deviations of the times (in seconds)of their four runners are as shown in the table: $$\begin{array} { c | c | c } 
\text { Runner } & \text { Mean } & \text { SD } \
\hline 1 & 12.19 & 0.12 \
2 & 12.37 & 0.14 \
3 & 12.01 & 0.08 \
4 & 11.73 & 0.07
\end{array}$$ What are the mean and standard deviation of the relay team's total time in this event? Assume that the runners' performances are independent.

Accepted Answer

A) ? = 48.3 sec,? = 12.64 sec 
B) ? = 48.3 sec,? = 0.045 sec 
C) ? = 24.15 sec,? = 0.21 sec 
D) ? = 48.3 sec,? = 0.21 sec 
E) ? = 24.15 sec,? = 12.64 sec

Question 10

A tennis player usually makes a successful first serve 69% of the time.She buys a new racket hoping that it will improve her success rate.When she first tests her new racket she makes 4 first serves in a row.Is this evidence that with the new racket her success rate has improved? In other words,is a streak like this unusual for her? Explain.

Accepted Answer

A) Yes; if her success rate were still 69%,she would have only a 0.92% chance of making 4 shots in a row.That's an unusual result. 
B) No; if her success rate were still 69%,she would have a 22.7% chance of making 4 shots in a row.That's not an unusual result. 
C) Yes; if her success rate were still 69%,she would have only a 2.8% chance of making 4 shots in a row.That's an unusual result. 
D) No; if her success rate were still 69%,she would have a 27.6% chance of making 4 shots in a row.That's not an unusual result. 
E) Yes; if her success rate were still 69%,she would have only a 22.7% chance of making 4 shots in a row.That's an unusual result.

Question 11

A carnival game offers a $80 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $5 to play and you're willing to spend up to $20 trying to win.You estimate that you have a 8% chance of hitting the balloon on any throw.Find the standard deviation of the number of darts you throw.Assume that throws are independent of each other.

Accepted Answer

A) 0.94 
B) 0.88 
C) 0.79 
D) 1.01 
E) 0.81

Question 12

A tennis player makes a successful first serve 54% of the time.If she serves 5 times,what is the probability that she gets exactly 3 first serves in? Assume that each serve is independent of the others.

Accepted Answer

A) 0.0333 
B) 0.3332 
C) 0.1575 
D) 0.2838 
E) 0.6668

Question 13

Jo is a hairstylist.The probability model below describes the number of clients that she may see in a day. $$\begin{array} { l | c c c c c c } 
\text { Number of clients } & 0 & 1 & 2 & 3 & 4 & 5 \
\hline \text { Probability } & 0.05 & 0.15 & 0.15 & 0.3 & 0.2 & 0.15
\end{array}$$ What is the standard deviation of the number of clients that Jo sees per day?

Accepted Answer

A) 1.31 
B) 1.41 
C) 2.19 
D) 1.99 
E) 1.55

Question 14

A hair salon claims that 90% of their customers are satisfied.However,Toni who recently started working there found that of 120 customers last week,only 100 were satisfied.If it were true that on average 90% of customers are satisfied,what is the chance that among 120 customers 100 or fewer would be satisfied?

Accepted Answer

A) 0.2297 
B) 0.9925 
C) 0.7703 
D) 0.0075 
E) 0.0003

Question 15

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 0.3Y. $$\begin{array} { c | c c } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 70 & 7 \
\mathrm { Y } & 60 & 8
\end{array}$$

Accepted Answer

A) ? = 18,? = 0.72 
B) ? = 60.3,? = 8 
C) ? = 60.3,? = 2.4 
D) ? = 18,? = 8 
E) ? = 18,? = 2.4

Question 16

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 no wrong numbers.

Accepted Answer

A) 0.9392 
B) 0.0608 
C) 0.1703 
D) 0.8297 
E) 0.0669

Question 17

Police estimate that in one city 48% of drivers wear their seat belts.They set up a safety roadblock,stopping cars to check for seat belt use.If they stop 70 cars during the first hour,what is the mean number of drivers expected to be wearing their seat belts?

Accepted Answer

A) 4.18 
B) 17.47 
C) 33.6 
D) 36.4 
E) 35

Question 18

In a box of 7 batteries,3 are dead.You choose two batteries at random from the box.Let the random variable X be the number of good batteries you get.Find the probability model for X.

Accepted Answer

A)  $$\begin{array} { l | c c c } 
\text { Number good } & 0 & 1 & 2 \
\hline \mathrm { P } \text { (Number good) } & 0.184 & 0.490 & 0.327
\end{array}$$ 
B)  $$\begin{array} { l | c c c } 
\text { Number good } & 0 & 1 & 2 \
\hline \mathrm { P } \text { (Number good) } & 0.143 & 0.571 & 0.286
\end{array}$$ 
C)  $$\begin{array} { l | c c c } 
\text { Number good } & 0 & 1 & 2 \
\hline \mathrm { P } \text { (Number good) } & 0.143 & 0.286 & 0.286
\end{array}$$ 
D)  $$\begin{array} { l | c c c } 
\text { Number good } & 0 & 1 & 2 \
\hline \mathrm { P } \text { (Number good) } & 0.286 & 0.571 & 0.286
\end{array}$$ 
E)  $$\begin{array} { l | c c c } 
\text { Number good } & 0 & 1 & 2 \
\hline \mathrm { P } \text { (Number good) } & 0.067 & 0.467 & 0.467
\end{array}$$

Question 19

Given independent random variables with means and standard deviations as shown,find the mean and standard deviation of the variable 3X - Y.Round to two decimal places if necessary. $$\begin{array} { l | l l } 
& \text { Mean } & \text { SD } \
\hline \mathrm { X } & 190 & 15 \
\mathrm { Y } & 170 & 19
\end{array}$$

Accepted Answer

A) ? = 60,? = 26 
B) ? = 740,? = 48.85 
C) ? = 400,? = 40.79 
D) ? = 400,? = 48.85 
E) ? = 400,? = 26

Question 20

A laboratory worker finds that 1.8% of his blood samples test positive for the HIV virus.If 270 blood samples are selected at random,is it appropriate to use a Normal model to approximate the distribution of the number that test positive for the HIV virus?

Accepted Answer

A) Yes; Normal model with μ = 4.86 and σ = 4.77 can be used to approximate the distribution. 
B) No; Normal model cannot be used to approximate the distribution because nq < 10. 
C) Yes; Normal model with μ = 4.86 and σ = 2.18 can be used to approximate the distribution. 
D) No; Normal model cannot be used to approximate the distribution because np < 10. 
E) Yes; Normal model with μ = 265.14 and σ = 2.18 can be used to approximate the distribution.

Exam 14: Random Variables