Statistics for Business and Economics: Global Edition

Newbold P., Carlson W.L., Thorne B.M.

Chapter 5

Continuous Probability Distributions - all with Video Answers

Educators

Chapter Questions

Problem 1

Using the uniform probability density function shown in Figure 5.7, find the probability that the random variable $X$ is between 1.4 and 1.8 .

Sheryl Ezze

Sheryl Ezze

Numerade Educator

Problem 2

Using the uniform probability density function shown in Figure 5.7, find the probability that the random variable $X$ is between 1.0 and 1.9 .

Sheryl Ezze

Numerade Educator

Problem 3

Using the uniform probability density function shown in Figure 5.7, find the probability that the random variable $X$ is less than 1.4 .

Sheryl Ezze

Numerade Educator

Problem 4

Using the uniform probability density function shown in Figure 5.7, find the probability that the random variable $X$ is greater than 1.3 .

Sheryl Ezze

Numerade Educator

Problem 5

An analyst has available two forecasts, $F_1$ and $F_2$ of earnings per share of a corporation next year. He intends to form a compromise forecast as a weighted average of the two individual forecasts. In forming the compromise forecast, weight $X$ will be given to the first forecast and weight $(1-X)$, to the second, so that the compromise forecast is $X F_1+(1-X) F_2$. The analyst wants to choose a value between 0 and 1 for the weight $X$, but he is quite uncertain of what will be the best choice. Suppose that what eventually emerges as the best possible choice of the weight $X$ can be viewed as a random variable uniformly distributed between 0 and 1 , having the probability density function
$$
f(x)= \begin{cases}1 & \text { for } 0 \leq x \leq 1 \\ 0 & \text { for all other } x\end{cases}
$$
a. Graph the probability density function.
b. Find and graph the cumulative distribution function.
c. Find the probability that the best choice of the weight $X$ is less than 0.25 .
d. Find the probability that the best choice of the weight $X$ is more than 0.75 .
e. Find the probability that the best choice of the weight $X$ is between 0.2 and 0.8 .

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 6

The jurisdiction of a rescue team includes emergencies occurring on a stretch of river that is 4 miles long. Experience has shown that the distance along this stretch, measured in miles from its northernmost point, at which an emergency occurs can be represented by a uniformly distributed random variable over the range 0 to 4 miles. Then, if $X$ denotes the distance (in miles) of an emergency from the northernmost point of this stretch of river, its probability density function is as follows:
$$
f(x)= \begin{cases}0.25 & \text { for } 0<x<4 \\ 0 & \text { for all other } x\end{cases}
$$
a. Graph the probability density function
b. Find and graph the cumulative distribution function.
c. Find the probability that a given emergency arises within 1 mile of the northernmost point of this stretch of river.
d. The rescue team's base is at the midpoint of this stretch of river. Find the probability that a given emergency arises more than 1.5 miles from this base.

Christopher Stanley

Christopher Stanley

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Problem 7

The incomes of all families in a particular suburb can be represented by a continuous random variable. It is known that the median income for all families in this suburb is $$\$ 60,000$$ and that $40 \%$ of all families in the suburb have incomes above $$\$ 72,000$$.
a. For a randomly chosen family, what is the probability that its income will be between $$\$ 60,000$$ and $$\$ 72,000$$ ?
b. Given no further information, what can be said about the probability that a randomly chosen family has an income below $$\$ 65,000$$ ?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 8

At the beginning of winter, a homeowner estimates that the probability is 0.4 that his total heating bill for the three winter months will be less than $$\$ 380$$. He also estimates that the probability is 0.6 that the total bill will be less than $$\$ 460$$.
a. What is the probability that the total bill will be between $$\$ 380$$ and $$\$ 460$$ ?
b. Given no further information, what can be said about the probability that the total bill will be less than $$\$ 400$$ ?

Xiaomeng Zhang

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Problem 9

The total cost for a production process is equal to $$\$ 1,000$$ plus two times the number of units produced. The mean and variance for the number of units produced are 500 and 900 , respectively. Find the mean and variance of the total cost.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 10

The profit for a production process is equal to $$\$ 1,000$$ minus two times the number of units produced. The mean and variance for the number of units produced are 50 and 90 , respectively. Find the mean and variance of the profit.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 11

The profit for a production process is equal to $$\$ 2,000$$ minus two times the number of units produced. The mean and variance for the number of units produced are 500 and 900 , respectively. Find the mean and variance of the profit.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 12

The profit for a production process is equal to $$\$ 6,000$$ minus three times the number of units produced. The mean and variance for the number of units produced are 1,000 and 900 , respectively. Find the mean and variance of the profit.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 13

An author receives a contract from a publisher, according to which she is to be paid a fixed sum of $$\$ 10,000$$ plus $$\$ 1.50$$ for each copy of her book sold. Her uncertainty about total sales of the book can be represented by a random variable with a mean of 30,000
and a standard deviation of 8,000 . Find the mean and standard deviation of the total payments she will receive.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 14

A contractor submits a bid on a project for which more research and development work needs to be done. It is estimated that the total cost of satisfying the project specifications will be $$\$ 20$$ million plus the cost of the further research and development work. The contractor views the cost of this additional work as a random variable with a mean of $$\$ 4$$ million and a standard deviation of $$\$ 1$$ million. The contractor wishes to submit a bid such that his expected profit will be $10 \%$ of his expected costs. What should be the bid? If this bid is accepted, what will be the standard deviation of the profit made by the project?

Sheryl Ezze

Numerade Educator

Problem 15

A charitable organization solicits donations by telephone. Employees are paid $$\$ 60$$ plus $20 \%$ of the money their calls generate each week. The amount of money generated in a week can be viewed as a random variable with a mean of $$\$ 700$$ and a standard deviation of $$\$ 130$$. Find the mean and standard deviation of an $\mathrm{em}$ ployee's total pay in a week.

Jen H

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Problem 16

A salesperson receives an annual salary of $$\$ 6,000$$ plus $8 \%$ of the value of the orders she takes. The annual value of these orders can be represented by a random variable with a mean of $$\$ 600,000$$ and a standard deviation of $$\$ 180,000$$. Find the mean and standard deviation of the salesperson's annual income.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 17

Let the random variable $\mathrm{Z}$ follow a standard normal distribution.
a. Find $P(Z<1.20)$.
b. Find $P(Z>1.33)$.
c. Find $P(Z>-1.70)$.
d. Find $P(Z>-1.00)$.
e. Find $P(1.20<Z<1.33)$.
f. Find $P(-1.70<Z<1.20)$.
g. Find $P(-1.70<Z<-1.00)$.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 18

Let the random variable $Z$ follow a standard normal distribution.
a. The probability is 0.70 that $Z$ is less than what number?
b. The probability is 0.25 that $Z$ is less than what number?
c. The probability is 0.2 that $\mathrm{Z}$ is greater than what number?
d. The probability is 0.6 that $\mathrm{Z}$ is greater than what number?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 19

Let the random variable $X$ follow a normal distribution with $\mu=50$ and $\sigma^2=64$.
a. Find the probability that $X$ is greater than 60 .
b. Find the probability that $X$ is greater than 35 and less than 62.
c. Find the probability that $X$ is less than 55.
d. The probability is 0.2 that $X$ is greater than what number?
e. The probability is 0.05 that $X$ is in the symmetric interval about the mean between which two numbers?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 20

Let the random variable $X$ follow a normal distribution with $\mu=80$ and $\sigma^2=100$.
a. Find the probability that $X$ is greater than 60.
b. Find the probability that $X$ is greater than 72 and less than 82 .
c. Find the probability that $X$ is less than 55 .
d. The probability is 0.1 that $X$ is greater than what number?
e. The probability is 0.6826 that $X$ is in the symmetric interval about the mean between which two numbers?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 21

Let the random variable $X$ follow a normal distribution with $\mu=0.2$ and $\sigma^2=0.0025$.
a. Find the probability that $X$ is greater than 0.4 .
b. Find the probability that $X$ is greater than 0.15 and less than 0.28 .
c. Find the probability that $X$ is less than 0.10 .
d. The probability is 0.2 that $X$ is greater than what number?
e. The probability is 0.05 that $X$ is in the symmetric interval about the mean between which two numbers?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 22

It is known that amounts of money spent on clothing in a year by students on a particular campus follow a normal distribution with a mean of $$\$ 380$$ and a standard deviation of $$\$ 50$$.
a. What is the probability that a randomly chosen student will spend less than $$\$ 400$$ on clothing in a year?
b. What is the probability that a randomly chosen student will spend more than $$\$ 360$$ on clothing in a year?
c. Draw a graph to illustrate why the answers to parts (a) and (b) are the same.
d. What is the probability that a randomly chosen student will spend between $$\$ 300$$ and $$\$ 400$$ on clothing in a year?
e. Compute a range of yearly clothing expendituresmeasured in dollars - that includes $80 \%$ of all students on this campus? Explain why any number of such ranges could be found, and find the shortest one.

Sheryl Ezze

Numerade Educator

Problem 23

Anticipated consumer demand in a restaurant for free-range steaks next month can be modeled by a normal random variable with mean 1,200 pounds and standard deviation 100 pounds.
a. What is the probability that demand will exceed 1,000 pounds?
b. What is the probability that demand will be between 1,100 and 1,300 pounds?
c. The probability is 0.10 that demand will be more than how many pounds?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 24

The tread life of Road Stone tires has a normal distribution with a mean of 35,000 miles and a standard deviation of 4,000 miles.
a. What proportion of these tires has a tread life of more than 38,000 miles?
b. What proportion of these tires has a tread life of less than 32,000 miles?
c. What proportion of these tires has a tread life of between 32,000 and 38,000 miles?
d. Draw a graph of the probability density function of tread lives, illustrating why the answers to parts (a) and (b) are the same and why the answers to parts (a), (b), and (c) sum to 1.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 25

An investment portfolio contains stocks of a large number of corporations. Over the last year the rates of return on these corporate stocks followed a normal distribution with mean $12.2 \%$ and standard deviation $7.2 \%$.
a. For what proportion of these corporations was the rate of return higher than $20 \%$ ?
b. For what proportion of these corporations was the rate of return negative?
c. For what proportion of these corporations was the rate of return between $5 \%$ and $15 \%$ ?

Danielle Fairburn

Danielle Fairburn

Numerade Educator

Problem 26

Southwest Co-op produces bags of fertilizer, and it is concerned about impurity content. It is believed that the weights of impurities per bag are normally distributed with a mean of 12.2 grams and a standard deviation of 2.8 grams. A bag is chosen at random.
a. What is the probability that it contains less than 10 grams of impurities?
b. What is the probability that it contains more than 15 grams of impurities?
c. What is the probability that it contains between 12 and 15 grams of impurities?
d. It is possible, without doing the detailed calculations, to deduce which of the answers to parts (a) and (b) will be the larger. How would you do this?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 27

A contractor has concluded from his experience that the cost of building a luxury home is a normally distributed random variable with a mean of $$\$ 500,000$$ and a standard deviation of $$\$ 50,000$$.
a. What is the probability that the cost of building a home will be between $$\$ 460,000$$ and $$\$ 540,000$$ ?
b. The probability is 0.2 that the cost of building will be less than what amount?
c. Find the shortest range such that the probability is 0.95 that the cost of a luxury home will fall in this range.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 28

Scores on an economics test follow a normal distribution. What is the probability that a randomly selected student will achieve a score that exceeds the mean score by more than 1.5 standard deviations?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 29

A new television series is to be shown. A broadcasting executive feels that his uncertainty about the rating that the show will receive in its first month can be represented by a normal distribution with a mean of 18.2 and a standard deviation of 1.5 . According to this executive, the probability is 0.1 that the rating will be less than what number?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 30

A broadcasting executive is reviewing the prospects for a new television series. According to his judgment, the probability is 0.25 that the show will achieve a rating higher than 17.8 , and the probability is 0.15 that it will achieve a rating higher than 19.2 . If the executive's uncertainty about the rating can be represented by a normal distribution, what are the mean and variance of that distribution?

Sheryl Ezze

Numerade Educator

Problem 31

The number of hits per day on the Web site of Professional Tool, Inc., is normally distributed with a mean of 700 and a standard deviation of 120 .
a. What proportion of days has more than 820 hits per day?
b. What proportion of days has between 730 and 820 hits?
c. Find the number of hits such that only $5 \%$ of the days will have the number of hits below this number.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 32

I am considering two alternative investments. In both cases I am unsure about the percentage return but believe that my uncertainty can be represented by normal distributions with the means and standard deviations shown in the accompanying table. I want to make the investment that is more likely to produce a return of at least $10 \%$. Which investment should I choose?
$$
\begin{array}{lcc}
\hline & \text { Mean } & \text { Standard Deviation } \\
\hline \text { Investment A } & 10.4 & 1.2 \\
\text { Investment B } & 11.0 & 4.0 \\
\hline
\end{array}
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 33

Tata Motors, Ltd., purchases computer process chips from two suppliers, and the company is concerned about the percentage of defective chips. A review of the records for each supplier indicates that the percentage defectives in consignments of chips follow normal distributions with the means and standard deviations given in the following table. The company is particularly anxious that the percentage of defectives in a consignment not exceed $5 \%$ and wants to purchase from the supplier that's more likely to meet that specification. Which supplier should be chosen?
$$
\begin{array}{lcc}
\hline & \text { Mean } & \text { Standard Deviation } \\
\hline \text { Supplier A } & 4.4 & 0.4 \\
\text { Supplier B } & 4.2 & 0.6 \\
\hline
\end{array}
$$

Sheryl Ezze

Numerade Educator

Problem 34

A furniture manufacturer has found that the time spent by workers assembling a particular table follows a normal distribution with a mean of 150 minutes and a standard deviation of 40 minutes.
a. The probability is 0.9 that a randomly chosen table requires more than how many minutes to assemble?
b. The probability is 0.8 that a randomly chosen table can be assembled in fewer than how many minutes?
c. Two tables are chosen at random. What is the probability that at least one of them requires at least 2 hours to assemble?

Rashmi Sinha

Numerade Educator

Problem 35

A company services copiers. A review of its records shows that the time taken for a service call can be represented by a normal random variable with a mean of 75 minutes and a standard deviation of 20 minutes.
a. What proportion of service calls takes less than 1 hour?
b. What proportion of service calls takes more than 90 minutes?
c. Sketch a graph to show why the answers to parts (a) and (b) are the same.
d. The probability is 0.1 that a service call takes more than how many minutes?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 36

Scores on an achievement test are known to be normally distributed with a mean of 420 and a standard deviation of 80 .
a. For a randomly chosen person taking this test, what is the probability of a score between 400 and 480 ?
b. What is the minimum test score needed in order to be in the top $10 \%$ of all people taking the test?
c. For a randomly chosen individual, state, without doing the calculations, in which of the following ranges his score is most likely to be: $400-439$, $440-479,480-519$, or $520-559$.
d. In which of the ranges listed in part (c) is the individual's score least likely to be?
e. Two people taking the test are chosen at random. What is the probability that at least one of them scores more than 500 points?

Sheryl Ezze

Numerade Educator

Problem 37

It is estimated that the time that a well-known rock band, the Living Ingrates, spends on stage at its concerts follows a normal distribution with a mean of 200 minutes and a standard deviation of 20 minutes.
a. What proportion of concerts played by this band lasts between 180 and 200 minutes?
b. An audience member smuggles a tape recorder into a Living Ingrates concert. The reel-to-reel tapes have a capacity of 245 minutes. What is the probability that this capacity will be insufficient to record the entire concert?
c. If the standard deviation of concert time was only 15 minutes, state, without doing the calculations, whether the probability that a concert would last more than 245 minutes would be larger than, smaller than, or the same as that found in part (b). Sketch a graph to illustrate your answer.
d. The probability is 0.1 that a Living Ingrates concert will last less than how many minutes? (Assume, as originally, that the population standard deviation is 20 minutes.)

Danielle Fairburn

Danielle Fairburn

Numerade Educator

Problem 38

The amount of time necessary for a student of statistics to solve assignments is, on average, 15 minutes. This can be modeled as a random normal variable with a standard deviation of 2 minutes. Calculate the probability that an assignment is instead solved between 14 and 16 minutes.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 39

Given a random sample size of $n=900$ from a binomial probability distribution with $P=0.50$ do the following:
a. Find the probability that the number of successes is greater than 500 .
b. Find the probability that the number of successes is fewer than 430 .
c. Find the probability that the number of successes is between 440 and 480 .
d. With probability 0.10 , the number of successes is fewer than how many?
e. With probability 0.08 , the number of successes is greater than how many?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 40

Given a random sample size of $n=1,600$ from a binomial probability distribution with $P=0.40$, do the following:
a. Find the probability that the number of successes is greater than 1,650 .
b. Find the probability that the number of successes is fewer than 1,530 .
c. Find the probability that the number of successes is between 1,550 and 1,650 .
d. With probability 0.09 , the number of successes is fewer than how many?
e. With probability 0.20 , the number of successes is greater than how many?

Sheryl Ezze

Numerade Educator

Problem 41

Given a random sample size of $n=900$ from a binomial probability distribution with $P=0.10$ do the following:
a. Find the probability that the number of successes is greater than 110.
b. Find the probability that the number of successes is fewer than 53.
c. Find the probability that the number of successes is between 55 and 120 .
d. With probability 0.10 , the number of successes is fewer than how many?
e. With probability 0.08 , the number of successes is greater than how many?

Sheryl Ezze

Numerade Educator

Problem 42

Given a random sample size of $n=1,600$ from a binomial probability distribution with $P=0.40$ do the following:
a. Find the probability that the percentage of successes is greater than 0.45 .
b. Find the probability that the percentage of successes is less than 0.35 .
c. Find the probability that the percentage of successes is between 0.37 and 0.44 .
d. With probability 0.20 , the percentage of successes is less than what percent?
e. With probability 0.09 , the percentage of successes is greater than what percent?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 43

Given a random sample size of $n=400$ from a binomial probability distribution with $P=0.20$ do the following:
a. Find the probability that the percentage of successes is greater than 0.25 .
b. Find the probability that the percentage of successes is less than 0.15 .
c. Find the probability that the percentage of successes is between 0.17 and 0.24 .
d. With probability 0.15 , the percentage of successes is less than what percent?
e. With probability 0.11 , the percentage of successes is greater than what percent?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 44

A car-rental company has determined that the probability a car will need service work in any given month is 0.2 . The company has 900 cars.
a. What is the probability that more than 200 cars will require service work in a particular month?
b. What is the probability that fewer than 175 cars will need service work in a given month?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 45

It is known that $10 \%$ of all the items produced by a particular manufacturing process are defective. From the very large output of a single day, 400 items are selected at random.
a. What is the probability that at least 35 of the selected items are defective?
b. What is the probability that between 40 and 50 of the selected items are defective?
c. What is the probability that between 34 and 48 of the selected items are defective?
d. Without doing the calculations, state which of the following ranges of defectives has the highest probability: 38-39, 40-41, 42-43, 44-45, or 46-47.

Sheryl Ezze

Numerade Educator

Problem 46

A random sample of 100 blue-collar employees at a large corporation are surveyed to assess their attitudes toward a proposed new work schedule. If $60 \%$ of all blue-collar employees at this corporation favor the new schedule, what is the probability that fewer than 50 in the random sample will be in favor?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 47

A hospital finds that $25 \%$ of its accounts are at least 1 month in arrears. A random sample of 450 accounts was taken.
a. What is the probability that fewer than 100 accounts in the sample were at least 1 month in arrears?
b. What is the probability that the number of accounts in the sample at least 1 month in arrears was between 120 and 150 (inclusive)?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 48

The tread life of Stone Soup tires can be modeled by a normal distribution with a mean of 35,000 miles and a standard deviation of 4,000 miles. A sample of 100 of these tires is taken. What is the probability that more than 25 of them have tread lives of more than 38,000 miles?

Sheryl Ezze

Numerade Educator

Problem 49

Bags of a chemical produced by a company have impurity weights that can be represented by a normal distribution with a mean of 12.2 grams and a standard deviation of $2.8 \mathrm{grams}$. A random sample of 400 of these bags is taken. What is the probability that at least 100 of them contain fewer than 10 grams of impurities?

Sheryl Ezze

Numerade Educator

Problem 50

Given an arrival process with $\lambda=1.0$, what is the probability that an arrival occurs in the first $t=2$ time units?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 51

Given an arrival process with $\lambda=8.0$, what is the probability that an arrival occurs in the first $t=7$ time units?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 52

Given an arrival process with $\lambda=5.0$, what is the probability that an arrival occurs after $t=7$ time units?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 53

Given an arrival process with $\lambda=5.0$, what is the probability that an arrival occurs after $t=5$ time units?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 54

Given an arrival process with $\lambda=3.0$, what is the probability that an arrival occurs in the first $t=2$ time units?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 55

A professor sees students during regular office hours. Time spent with students follows an exponential distribution with a mean of 10 minutes.
a. Find the probability that a given student spends fewer than 20 minutes with the professor.
b. Find the probability that a given student spends more than 5 minutes with the professor.
c. Find the probability that a given student spends between 10 and 15 minutes with the professor.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 56

Times to gather preliminary information from arrivals at an outpatient clinic follow an exponential distribution with mean 15 minutes. Find the probability, for a randomly chosen arrival, that more than 18 minutes will be required.

Rashmi Sinha

Numerade Educator

Problem 57

It is known that for a laboratory computing system the number of system failures during a month has a Poisson distribution with a mean of 0.8 . The system has just failed. Find the probability that at least 2 months will elapse before a further failure.

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 58

Suppose that the time between successive occurrences of an event follows an exponential distribution with a mean of $1 / \lambda$ minutes. Assume that an event occurs.
a. Show that the probability that more than 3 minutes elapses before the occurrence of the next event is $e^{-3 \lambda}$.
b. Show that the probability that more than $6 \mathrm{~min}$ utes elapses before the occurrence of the next event is $e^{-6 \lambda}$.
c. Using the results of parts (a) and (b), show that if 3 minutes have already elapsed, the probability that a further 3 minutes will elapse before the next occurrence is $e^{-3 \lambda}$. Explain your answer in words.

Michael Nartey

Numerade Educator

Problem 59

A Lumix Panasonic camera has a rechargeable battery. The battery life before recharging is needed can be modeled as an exponential distribution with $\lambda=0.05$.
a. Calculate the standard deviation of the battery's life before recharging.
b. Calculate the probability that the battery will last more than 20 hours.

Michael Nartey

Numerade Educator

Problem 60

Delivery trucks arrive independently at the Floorstore Regional distribution center with various consumer items from the company's suppliers. The mean number of trucks arriving per hour is 20 . Given that a truck has just arrived answer the following:
a. What is the probability that the next truck will not arrive for at least 5 minutes?
b. What is the probability that the next truck will arrive within the next 2 minutes?
c. What is the probability that the next truck will arrive between 4 and 10 minutes?

Sheryl Ezze

Numerade Educator

Problem 61

A random variable $X$ is normally distributed with a mean of 100 and a variance of 100 , and a random variable $Y$ is normally distributed with a mean of 200 and a variance of 400 . The random variables have a correlation coefficient equal to 0.5 . Find the mean and variance of the random variable:
$$
W=5 X+4 Y
$$

Sheryl Ezze

Numerade Educator

Problem 62

A random variable $X$ is normally distributed with a mean of 100 and a variance of 100 , and a random variable $Y$ is normally distributed with a mean of 200 and a variance of 400 . The random variables have a correlation coefficient equal to -0.5 . Find the mean and variance of the random variable:
$$
W=5 X+4 Y
$$

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 63

A random variable $X$ is normally distributed with a mean of 100 and a variance of 100 , and a random variable $Y$ is normally distributed with a mean of 200 and a variance of 400 . The random variables have a correlation coefficient equal to 0.5 . Find the mean and variance of the random variable:
$$
W=5 X-4 Y
$$

Sheryl Ezze

Numerade Educator

Problem 64

A random variable $X$ is normally distributed with a mean of 500 and a variance of 100 , and a random variable $Y$ is normally distributed with a mean of 200 and a variance of 400 . The random variables have a correlation coefficient equal to 0.5 . Find the mean and variance of the random variable:
$$
W=5 X-4 Y
$$

Sheryl Ezze

Numerade Educator

Problem 65

A random variable $X$ is normally distributed with a mean of 100 and a variance of 500 , and a random variable $Y$ is normally distributed with a mean of 200 and a variance of 400 . The random variables have a correlation coefficient equal to -0.5 . Find the mean and variance of the random variable:
$$
W=5 X-4 Y
$$
Application Exercises

Sheryl Ezze

Numerade Educator

Problem 66

An investor plans to divide $$\$ 200,000$$ between two investments. The first yields a certain profit of $10 \%$, whereas the second yields a profit with expected value $18 \%$ and standard deviation $6 \%$. If the investor divides the money equally between these two investments, find the mean and standard deviation of the total profit.

Sheryl Ezze

Numerade Educator

Problem 67

A homeowner has installed a new energy-efficient furnace. It is estimated that over a year the new furnace will reduce energy costs by an amount that can be regarded as a random variable with a mean of $$\$ 200$$ and a standard deviation of $$\$ 60$$. Stating any assumptions you need to make, find the mean and standard deviation of the total energy cost reductions over a period of 5 years.

Sheryl Ezze

Numerade Educator

Problem 68

A consultant is beginning work on three projects. The expected profits from these projects are $$\$ 50,000$$, $$\$ 72,000$$, and $$\$ 40,000$$. The associated standard deviations are $$\$ 10,000, \$ 12,000$$, and $$\$ 9,000$$. Assuming independence of outcomes, find the mean and standard deviation of the consultant's total profit from these three projects.

Sheryl Ezze

Numerade Educator

Problem 69

A consultant has three sources of income-from teaching short courses, from selling computer software, and from advising on projects. His expected annual incomes from these sources are $$\$ 20,000, \$ 25,000$$, and $$\$ 15,000$$, and the respective standard deviations are $$\$ 2,000, \$ 5,000$$, and $$\$ 4,000$$. Assuming independence, find the mean and standard deviation of his total annual income.

Tanvi Garg

Numerade Educator

Problem 70

Five inspectors are employed to check the quality of components produced on an assembly line. For each inspector the number of components that can be checked in a shift can be represented by a random variable with mean 120 and standard deviation 15 . Let $X$ represent the number of components checked by an inspector in a shift. Then the total number checked is $5 X$, which has a mean of 600 and a standard deviation of 80 . What is wrong with this argument? Assuming that inspectors' performances are independent of one another, find the mean and standard deviation of the total number of components checked in a shift.

Sheryl Ezze

Numerade Educator

Problem 71

It is estimated that in normal highway driving, the number of miles that can be covered by automobiles of a particular model on 1 gallon of gasoline can be represented by a random variable with mean 28 and standard deviation 2.4. Sixteen of these cars, each with 1 gallon of gasoline, are driven independently under highway conditions. Find the mean and standard deviation of the average number of miles that will be achieved by these cars.

Sheryl Ezze

Numerade Educator

Problem 72

Shirley Johnson, portfolio manager, has asked you to analyze a newly acquired portfolio to determine its mean value and variability. The portfolio consists of 50 shares of Xylophone Music and 40 shares of Yankee Workshop. Analysis of past history indicates that the share price of Xylophone Music has a mean of 25 and a variance of 121 . A similar analysis indicates that Yankee has a mean share price of 40 with a variance of 225. Your best evidence indicates that the share prices have a correlation of +0.5 .
a. Compute the mean and variance of the portfolio.
b. Suppose that the correlation between share prices was actually -0.5 . Now what are the mean and variance of the portfolio?

Sheryl Ezze

Numerade Educator

Problem 73

Prairie Flower Cereal has annual sales revenue of $$\$ 400,000,000$$. George Severn, a 58-year-old senior vice president, is responsible for production and sales of Nougy 93 Fruity cereal. Daily production in cases is normally distributed, with a mean of 100 and a variance of 625. Daily sales in cases are also normally distributed, with a mean of 100 and a standard deviation of 8. Sales and production have a correlation of 0.60 . The selling price per case is $$\$ 10$$. The variable production cost per case is $$\$ 7$$. The fixed production costs per day are $$\$ 250$$.
a. What is the probability that total revenue is greater than total costs on any day?
b. Construct a $95 \%$ acceptance interval for total sales revenue minus total costs.

Barsha Rana

Numerade Educator

Problem 74

The nation of Olecarl, located in the South Pacific, has asked you to analyze international trade patterns. You first discover that each year it exports 10 units and imports 10 units of wonderful stuff. The price of exports is a random variable with a mean of 100 and a variance of 100 . The price of imports is a random variable with a mean of 90 and a variance of 400 . In addition, you discover that the prices of imports and exports have a correlation of $\rho=-0.40$. The prices of both exports and imports follow a normal probability density function. Define the balance of trade as the difference between the total revenue from exports and the total cost of imports.
a. What are the mean and variance of the balance of trade?
b. What is the probability that the balance of trade is negative?

Sheryl Ezze

Numerade Educator

Problem 75

You have been asked to determine the probability that the contribution margin for a particular product line exceeds the fixed cost of $$\$ 2,000$$. The total number of units sold is a normally distributed random variable with a mean of 400 and a variance of 900 , $X \sim N(400,900)$. The selling price per unit is $$\$ 10$$. The total number of units produced is a normally distributed random variable with a mean of 400 and a variance of $1,600, Y \sim N(400,1,600)$. The variable production cost is $$\$ 4$$ per unit. Production and sales have a positive correlation of 0.50 .

Sheryl Ezze

Numerade Educator

Problem 76

The nation of Waipo has recently created an economic development plan that includes expanded exports and imports. It has completed a series of extensive studies of the world economy and Waipo's economic capability, following Waipo's extensive 10-year educational-enhancement program. The resulting model indicates that in the next year exports will be normally distributed with a mean of 100 and a variance of 900 (in billions of Waipo yuan). In addition, imports are expected to be normally distributed with a mean of 105 and a variance of 625 in the same units. The correlation between exports and imports is expected to be +0.70 . Define the trade balance as exports minus imports.
a. Determine the mean and variance of the trade balance (exports minus imports) if the model parameters given above are true.
b. What is the probability that the trade balance will be positive?

Sheryl Ezze

Numerade Educator

Problem 77

A consultant knows that it will cost him $$\$ 10,000$$ to fulfill a particular contract. The contract is to be put out for bids, and he believes that the lowest bid, excluding his own, can be represented by a distribution that is uniform between $$\$ 8,000$$ and $$\$ 20,000$$. Therefore, if the random variable $\mathrm{X}$ denotes the lowest of all other bids
(in thousands of dollars), its probability density function is as follows:
$$
f(x)= \begin{cases}1 / 12 & \text { for } 8<x<20 \\ 0 & \text { for all other values of } x\end{cases}
$$
a. What is the probability that the lowest of the other bids will be less than the consultant's cost estimate of $$\$ 10,000$$ ?
b. If the consultant submits a bid of $$\$ 12,000$$, what is the probability that he will secure the contract?
c. The consultant decides to submit a bid of $$\$ 12,000$$. What is his expected profit from this strategy?
d. If the consultant wants to submit a bid so that his expected profit is as high as possible, discuss how he should go about making this choice.

Rashmi Sinha

Numerade Educator

Problem 78

The ages of a group of executives attending a convention are uniformly distributed between 35 and 65 years. If the random variable $X$ denotes ages in years, the probability density function is as follows:
$$
f(x)= \begin{cases}1 / 30 & \text { for } 35<x<65 \\ 0 & \text { for all other values of } x\end{cases}
$$
a. Graph the probability density function for $X$.
b. Find and graph the cumulative distribution function for $X$.
c. Find the probability that the age of a randomly chosen executive in this group is between 40 and 50 years.
d. Find the mean age of the executives in the group.

Sheryl Ezze

Numerade Educator

Problem 79

The random variable $X$ has probability density function as follows:
$$
f(x)= \begin{cases}x & \text { for } 0<x<1 \\ 2-x & \text { for } 1<x<2 \\ 0 & \text { for all other values of } x\end{cases}
$$
a. Graph the probability density function for $X$.
b. Show that the density has the properties of a proper probability density function.
c. Find the probability that $X$ takes a value between 0.5 and 1.5 .

Sheryl Ezze

Numerade Educator

Problem 80

An investor puts $$\$ 2,000$$ into a deposit account with a fixed rate of return of $10 \%$ per year. A second sum of $$\$ 1,000$$ is invested in a fund with an expected rate of return of $16 \%$ and a standard deviation of $8 \%$ per year.
a. Find the expected value of the total amount of money this investor will have after a year.
b. Find the standard deviation of the total amount after a year.

Sheryl Ezze

Numerade Educator

Problem 81

A hamburger stand sells hamburgers for $$\$ 1.45$$ each. Daily sales have a distribution with a mean of 530 and a standard deviation of 69.
a. Find the mean daily total revenues from the sale of hamburgers.
b. Find the standard deviation of total revenues from the sale of hamburgers.
c. Daily costs (in dollars) are given by
$$
C=100+0.95 X
$$
where $X$ is the number of hamburgers sold. Find the mean and standard deviation of daily profits from sales.

Sheryl Ezze

Numerade Educator

Problem 82

An analyst forecasts corporate earnings, and her record is evaluated by comparing actual earnings with predicted earnings. Define the following:
actual earnings $=$ predicted earnings + forecast error
If the predicted earnings and forecast error are independent of each other, show that the variance of predicted earnings is less than the variance of actual earnings.

Sheryl Ezze

Numerade Educator

Problem 83

Let $X_1$ and $X_2$ be a pair of random variables. Show that the covariance between the random variables $Y_1=\left(X_1+X_2\right)$ and $Y_2=\left(X_1-X_2\right)$ is 0 if and only if $X_1$ and $X_2$ have the same variance.

Ameer Said

Numerade Educator

Problem 84

Grade point averages of students on a large campus follow a normal distribution with a mean of 2.6 and a standard deviation of 0.5 .
a. One student is chosen at random from this campus. What is the probability that this student has a grade point average higher than 3.0 ?
b. One student is chosen at random from this campus. What is the probability that this student has a grade point average between 2.25 and 2.75 ?
c. What is the minimum grade point average needed for a student's grade point average to be among the highest $10 \%$ on this campus?
d. A random sample of 400 students is chosen from this campus. What is the probability that at least 80 of these students have grade point averages higher than 3.0 ?
e. Two students are chosen at random from this campus. What is the probability that at least one of them has a grade point average higher than 3.0 ?

Sheryl Ezze

Numerade Educator

Problem 85

A company services home air conditioners. It is known that times for service calls follow a normal distribution with a mean of 60 minutes and a standard deviation of 10 minutes.
a. What is the probability that a single service call takes more than 65 minutes?
b. What is the probability that a single service call takes between 50 and 70 minutes?
c. The probability is 0.025 that a single service call takes more than how many minutes?
d. Find the shortest range of times that includes $50 \%$ of all service calls.
e. A random sample of four service calls is taken. What is the probability that exactly two of them take more than 65 minutes?

Sheryl Ezze

Numerade Educator

Problem 86

It has been found that times taken by people to complete a particular tax form follow a normal distribution with a mean of 100 minutes and a standard deviation of 30 minutes.
a. What is the probability that a randomly chosen person takes less than 85 minutes to complete this form?
b. What is the probability that a randomly chosen person takes between 70 and 130 minutes to complete this form?
c. Five percent of all people take more than how many minutes to complete this form?
d. Two people are chosen at random. What is the probability that at least one of them takes more than an hour to complete this form?
e. Four people are chosen at random. What is the probability that exactly two of them take longer than an hour to complete this form?
f. For a randomly chosen person, state in which of the following ranges (expressed in minutes) the time to complete the form is most likely to lie.
$70-89,90-109,100-129, \quad 130-149$
g. For a randomly chosen person, state in which of the following ranges (expressed in minutes) the time to complete the form is least likely to lie.
$70-89, \quad 90-109, \quad 110-129, \quad 130-149$

Sheryl Ezze

Numerade Educator

Problem 87

A pizza delivery service delivers to a campus dormitory. Delivery times follow a normal distribution with a mean of 20 minutes and a standard deviation of 4 minutes.
a. What is the probability that a delivery will take between 15 and 25 minutes?
b. The service does not charge for the pizza if delivery takes more than 30 minutes. What is the probability of getting a free pizza from a single order?
c. During final exams, a student plans to order pizza five consecutive evenings. Assume that these delivery times are independent of each other. What is the probability that the student will get at least one free pizza?
d. Find the shortest range of times that includes $40 \%$ of all deliveries from this service.
e. For a single delivery, state in which of the following ranges (expressed in minutes) the delivery time is most likely to lie.
$$
18-20, \quad 19-21, \quad 20-22, \quad 21-23
$$
f. For a single delivery, state in which of the following ranges (expressed in minutes) the delivery time is least likely to lie.
$$
18-20, \quad 19-21, \quad 20-22, \quad 21-23
$$

Sheryl Ezze

Numerade Educator

Problem 88

A video-rental chain estimates that annual expenditures of members on rentals follow a normal distribution with a mean of $$\$ 100$$. It was also found that $10 \%$ of all members spend more than $$\$ 130$$ in a year. What percentage of members spends more than $$\$ 140$$ in a year?

Sheryl Ezze

Numerade Educator

Problem 89

It is estimated that amounts of money spent on gasoline by customers at a gas station follow a normal distribution with a standard deviation of $$\$ 2.50$$. It is also found that $10 \%$ of all customers spent more than $$\$ 25$$. What percentage of customers spent less than $$\$ 20$$ ?

Sheryl Ezze

Numerade Educator

Problem 90

A market research organization has found that $40 \%$ of all supermarket shoppers refuse to cooperate when questioned by its pollsters. If 1,000 shoppers are approached, what is the probability that fewer than 500 will refuse to cooperate?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 91

An organization that gives regular seminars on sales motivation methods determines that $60 \%$ of its clients have attended previous seminars. From a sample of 400 clients what is the probability that more than half have attended previous seminars?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 92

An ambulance service receives an average of 15 calls per day during the time period 6 p.m. to 6 a.m. for assistance. For any given day what is the probability that fewer than 10 calls will be received during the 12 -hour period? What is the probability that more than 17 calls during the 12 -hour period will be received?

Sheryl Ezze

Numerade Educator

Problem 93

In a large department store a customer-complaints office handles an average of six complaints per hour about the quality of service. The distribution is Poisson.
a. What is the probability that in any hour exactly six complaints will be received?
b. What is the probability that more than 20 minutes will elapse between successive complaints?
c. What is the probability that fewer than 5 minutes will elapse between successive complaints?
d. The store manager observes the complaints office for a 30-minute period, during which no complaints are received. He concludes that a talk he gave to his staff on the theme "the customer is always right" has obviously had a beneficial effect. Suppose that, in fact, the talk had no effect. What is the probability of the manager observing the office for a period of 30 minutes or longer with no complaints?

Hoan Nguyen

Numerade Educator

Problem 94

A fish market in Hong Kong offers a large variety of fresh fish on its stands. You have found out that the average chunk of tuna sushi on sale has a weight of 3.2 grams, with a standard deviation of $0.8 \mathrm{gram}$. Assuming the weights of tuna sushi are normally distributed, what is the probability that a randomly selected piece of sushi will weigh more than 4.4 grams?

Christopher Stanley

Christopher Stanley

Numerade Educator

Problem 95

In a Godiva Chocolate Shop, there are different sizes and weights of boxes of truffles.
a. Find the probability that a box of truffles weighs between 283 and 285.4 grams. The mean weight of a box is 283 grams and the standard deviation is 1.6 grams.
b. After a more careful check, the standard deviation was found to be 2.2 grams. Find the new probability.

Sheryl Ezze

Numerade Educator

Problem 96

A management consultant found that the amount of time per day spent by executives performing tasks that could be done equally well by subordinates followed a normal distribution with a mean of 2.4 hours. It was also found that $10 \%$ of executives spent over 3.5 hours per day on tasks of this type. For a random sample of 400 executives, find the probability that more than 80 spend more than 3 hours per day on tasks of this type.

Sheryl Ezze

Numerade Educator

Problem 97

Financial Managers, Inc., buys and sells a large number of stocks routinely for the various accounts that it manages. Portfolio manager Andrea Colson has asked for your assistance in the analysis of the Johnson Fund. A portion of this portfolio consists of 10 shares of stock A and 8 shares of stock B. The price of A has a mean of 10 and a variance of 16 , while the price of $B$ has a mean of 12 and a variance of 9 . The correlation between prices is 0.3 .
a. What are the mean and variance of the portfolio value?
b. Andrea has been asked to reduce the variance (risk) of the portfolio. She offers to trade the 10 shares of stock $A$ and receives two offers, from which she can select one: 10 shares of stock 1 with a mean price of 10 , a variance of 25 , and a correlation with the price of stock $B$ equal to -0.2 ; or 10 shares of stock 2 with a mean price of 10 , a variance of 9 , and a correlation with the price of stock B equal to +0.5 . Which offer should she select?

Sheryl Ezze

Numerade Educator

Problem 98

Financial Managers, Inc., buys and sells a large number of stocks routinely for the various accounts that it manages. Portfolio manager Sarah Bloom has asked for your assistance in the analysis of the Burde Fund. A portion of this portfolio consists of 10 shares of stock $A$ and 8 shares of stock B. The price of $A$ has a mean of 12 and a variance of 14 , while the price of B has a mean of 10 and a variance of 12 . The correlation between prices is 0.5 .
a. What are the mean and variance of the portfolio value?
b. Sarah has been asked to reduce the variance (risk) of the portfolio. She offers to trade the 10 shares of stock $A$ and receives two offers from which she can select one: 10 shares of stock 1 with a mean price of 12 , a variance of 25 , and a correlation with the price of stock B equal to -0.2 ; or 10 shares of stock 2 with a mean price of 10 , a variance of 9 , and a correlation with the price of stock B, equal to +0.5 . Which offer should she select?

Sheryl Ezze

Numerade Educator

Problem 99

Big Nail Construction Inc. is building a large, new student center for a famous Midwestern liberal arts college. During the project Christine Buildumbig, the project manager, requests that a pile of sand weighing between 138,000 pounds and 141,000 pounds be placed on the newly constructed driveway. You have been asked to determine the probability that the delivered sand satisfies Christine's request. You have ordered that one big truck and one small truck be used to deliver the sand. Sand loads in the big truck are normally distributed with a mean of 80,000 and a variance of $1,000,000$, and sand loads in the small truck are also normally distributed with a mean weight of 60,000 pounds and a variance of 810,000 . From past experience with the sand-loading facility, you know that the weight of sand in the two trucks has a correlation of 0.40 . What is the probability that the resulting pile of sand has a weight that is between 138,000 and 141,000 pounds?

Sheryl Ezze

Numerade Educator

Problem 100

An investment portfolio in Singapore specializes in airline stocks and contains two of them. One is Singapore Airlines (mean: 0.12 ; standard deviation: 0.02 ), and it accounts for $30 \%$ of the portfolio shares. The other airline present in the portfolio is AirAsia (mean: 0.25 ; standard deviation: 0.15 ), a higher-risk, higherreturn investment.
a. What is the expected value and the standard deviation of the portfolio if the coefficient of correlation of the two stocks is 0.5 ?
b. What will they be if the correlation is 0.2 instead?

Sheryl Ezze

Numerade Educator

Problem 101

Shirley Johnson is developing a new mutual fund portfolio and in the process has asked you to develop the mean and variance for the stock price that consists of 10 shares of stocks from each of the following firms: Alcoa Inc, Reliant Energy, and Sea Container. Using the data file Stock Price File, compute the mean and variance for this portfolio. Prepare the analysis by using means, variances, and covariances for individual stocks following the methods used in Examples 5.16 and 5.17 then confirm your results by obtaining the portfolio price for each year using the computer. Assuming that the portfolio price is normally distributed, determine the narrowest interval that contains $95 \%$ of the distribution of portfolio value.

Sheryl Ezze

Numerade Educator

Problem 102

Zafer Toprak is a developing a new mutual fund portfolio and in the process has asked you to develop the mean and variance for the stock price that consists of 10 shares of stocks from Alcoa Inc., 20 shares from AB Volvo, 10 shares from TCF Financial, and 20 shares from Pentair Inc. Using the data file Stock Price File, compute the mean and variance for this portfolio. Prepare the analysis by using means, variances, and covariances for individual stocks following the methods used in Examples 5.16 and 5.17, and then confirm your results by obtaining the portfolio price for each year using the computer. Assuming that the portfolio price is normally distributed, determine the narrowest interval that contains $95 \%$ of the distribution of portfolio value.

Sheryl Ezze

Numerade Educator

Problem 103

Charles Thorson has asked you to determine the mean and variance for a portfolio that consists of 100 shares of stock from each of the following firms: $3 \mathrm{M}$ Company, Alcoa, Inc, Intel Corporation, Potlatch Corp., General Motors, and Sea Containers. Using the data file Stock Price File, compute the mean and variance for this portfolio. Assuming that the portfolio price is normally distributed determine the narrowest interval that contains $95 \%$ of the distribution of portfolio value.

Sheryl Ezze

Numerade Educator

Problem 104

You have been asked to evaluate the monthly stock price growth for a portfolio which contains the following firms: $3 \mathrm{M}$ Company, Alcoa, Inc., Intel Corporation, Potlatch Corp., General Motors, and Sea Containers. The fraction of the portfolio dollar value for each firm will be the same. Using the data file Return on Stock Price 60 month, compute the mean and variance for the stock price growth and the covariance between them. Then determine the mean and variance for the entire portfolio.

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Problem 105

Deep Water Financial of Duluth, Minnesota, has asked you to evaluate the stock price growth for a portfolio containing the following firms: General Motors, International Business Machines, Potlatch, Inc., Sea Containers, Ltd., and Tata Communications. Compute the means, variances, and covariances for the stocks. Using the data file Stock Price File, compute the mean and variance for a portfolio that represents the five stocks equally. Second, modify the portfolio by removing Potlatch and Sea Containers and including in the portfolio $40 \%$ General Motors, $30 \%$ International Business Machines, and 30\% Tata Communications. Determine the mean and variance for the second portfolio and compare it with the first.

Sheryl Ezze

Numerade Educator

Problem 106

Consider a portfolio that contains stocks from the following firms: AB Volvo, Pentair, Inc., Reliant Energy, Inc., TCF Financial, 3M Company, and Restoration Hardware. Data for these stocks for a 60 -month period (May 2003-April 2008) are contained in the data file Return on Stock Price 60 month. Compute the means, variances, and covariances for the monthly stock price growth rate. Determine the mean and variance for a portfolio that contains equal fractions of the six stocks. Construct a second portfolio by removing TCF Financial and Restoration Hardware. Determine the mean and variance of this second portfolio that includes $20 \%$ AB Volvo, $30 \%$ Pentair, 30\% Reliant Energy, and $20 \% 3 \mathrm{M}$ Company. Compare this portfolio with the first and recommend a choice between them.

Sheryl Ezze

Numerade Educator