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Essentials of Statistics for Business & Economics

David R. Anderson; Dennis J. Sweeney; Thomas A. Williams; Jeffrey D. Camm; James J. Cochran

Chapter 5

Discrete Probability Distributions - all with Video Answers

Educators

Chapter Questions

02:51

Problem 1

Consider the experiment of tossing a coin twice.
a. List the experimental outcomes.
b. Define a random variable that represents the number of heads occurring on the two tosses.
c. Show what value the random variable would assume for each of the experimental outcomes.
d. Is this random variable discrete or continuous?

Ahmad Reda
Ahmad Reda
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01:40

Problem 2

Consider the experiment of a worker assembling a product.
a. Define a random variable that represents the time in minutes required to assemble the product.
b. What values may the random variable assume?
c. Is the random variable discrete or continuous?

Dominador Tan
Dominador Tan
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01:48

Problem 3

Interviews at Brookwood Institute. Three students scheduled interviews for summer employment at the Brookwood Institute. In each case the interview results in either an offer for a position or no offer. Experimental outcomes are defined in terms of the results of the three interviews.
a. List the experimental outcomes.
b. Define a random variable that represents the number of offers made. Is the random variable continuous?
c. Show the value of the random variable for each of the experimental outcomes.

Nick Johnson
Nick Johnson
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01:00

Problem 4

Unemployment in Northeastern States. The Census Burcau includes nine states in what it defines as the Northeast region of the United States. Assume that the government is interested in tracking unemployment in these nine states and that the random variable of interest is the number of Northeastern states with an unemployment rate that is less than $8.3 \%$. What values may this random variable assume?

Nick Johnson
Nick Johnson
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02:42

Problem 5

Blood Test Analysis. To perform a certain type of blood analysis, lab technicians must perform two procedures. The first procedure requires either one or two separate steps, and the second procedure requires either one, two, or three steps.
a. List the experimental outcomes associated with performing the blood analysis.
b. If the random variable of interest is the total number of steps required to do the complete analysis (both procedures), show what value the random variable will assume for each of the experimental outcomes.

Lucas Finney
Lucas Finney
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02:41

Problem 6

Types of Random Variables. Listed below is a series of experiments and associated random variables. In each case, identify the values that the random variable can assume and state whether the random variable is discrete or continuous.

Nick Johnson
Nick Johnson
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00:57

Problem 7

The probability distribution for the random variable x follows.

$$
\begin{array}{rr}
\boldsymbol{x} & \boldsymbol{f ( x )} \\
20 & .20 \\
25 & .15 \\
30 & .25 \\
35 & 40
\end{array}
$$

a. Is this probability distribution valid? Explain.
b. What is the probability that $x=30$ ?
c. What is the probability that $x$ is less than or equal to 25 ?
d. What is the probability that $x$ is greater than 30 ?

Nick Johnson
Nick Johnson
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03:25

Problem 8

Operating Room Use. The following data were collected by counting the number of operating rooms in use at Tampa General Hospital over a 20-day period: On three of the days only one operating room was used, on five of the days two were used, on eight of the days three were used, and on four days all four of the hospital's operating rooms were used.
a. Use the relative frequency approach to construct an empirical discrete probability distribution for the number of operating rooms in use on any given day.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfies the required conditions for a valid discrete probability distribution.

Nick Johnson
Nick Johnson
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02:54

Problem 9

Employee Retention. Employee retention is a major concern for many companies. A survey of Americans asked how long they have worked for their current employer (Bureau of Labor Statistics website). Consider the following example of sample data of 2000 college graduates who graduated five years ago.

$$
\begin{array}{cc}
\text { Time with Current } & \\
\text { Employer (years) } & \text { Number } \\
1 & 506 \\
2 & 390 \\
3 & 310 \\
4 & 218 \\
5 & 576
\end{array}
$$

Let $x$ be the random variable indicating the number of years the respondent has worked for her/his current employer.
a. Use the data to develop an empirical discrete probability distribution for $x$.
b. Show that your probability distribution satisfies the conditions for a valid discrete probability distribution.
c. What is the probability that a respondent has been at her/his current place of employment for more than 3 years?

Nick Johnson
Nick Johnson
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04:36

Problem 10

Job Satisfaction of IS Managers. The percent frequency distributions of job satisfaction scores for a sample of information systems (IS) senior executives and middle managers are as follows. The scores range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied).

$$
\begin{array}{ccc}
\begin{array}{ccc}
\text { Job Satisfaction } \\
\text { Score }
\end{array} & \begin{array}{c}
\text { IS Senior } \\
\text { Executives (\%) }
\end{array} & \begin{array}{l}
\text { IS Middle } \\
\text { Managers (\%) }
\end{array} \\
1 & 5 & 4 \\
2 & 9 & 10 \\
3 & 3 & 12 \\
4 & 42 & 46 \\
5 & 41 & 28
\end{array}
$$

a. Develop a probability distribution for the job satisfaction score of a senior executive.
b. Develop a probability distribution for the job satisfaction score of a middle manager.
c. What is the probability a senior executive will report a job satisfaction score of 4 or 5 ?
d. What is the probability a middle manager is very satisfied?
c. Compare the overall job satisfaction of senior executives and middle managers.

Michael Nartey
Michael Nartey
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01:30

Problem 11

Mailing Machine Malfunctions. A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take 1, 2, 3, or 4 hours. The different types of malfunctions occur at about the same frequency.
a. Develop a probability distribution for the duration of a service call.
b. Draw a graph of the probability distribution.
c. Show that your probability distribution satisfies the conditions required for a discrete probability function.
d. What is the probability a service call will take three hours?
e. A service call has just come in, but the type of malfunction is unknown. It is 3:00 P.M. and service technicians usually get off at $5 ; 00 \mathrm{P} . \mathrm{M}$. What is the probability the service technician will have to work overtime to fix the machine today?

Nick Johnson
Nick Johnson
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02:04

Problem 12

New Cable Subseribers. Spectrum provides cable television and Internet service to millions of customers. Suppose that the management of Spectrum subjectively assesses a probability distribution for the number of new subscribers next year in the state of New York as follows.
$$
\begin{array}{cc}
\boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\
100,000 & 10 \\
200,000 & 20 \\
300,000 & 25 \\
400,000 & .30 \\
500,000 & .10 \\
600,000 & .05
\end{array}
$$

a. Is this probability distribution valid? Explain.
b. What is the probability Spectrum will obtain more than 400,000 new subscribers?
c. What is the probability Spectrum will obtain fewer than 200,000 new subscribers?

Nick Johnson
Nick Johnson
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02:34

Problem 13

Establishing Patient Trust. A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1,2 , or 3 . Let $x$ be a random variable indicating the number of sessions required to gain the patient's trust. The following probability function has been proposed.
$$
f(x)=\frac{x}{6} \quad \text { for } x=1,2, \text { or } 3
$$
a. Is this probability function valid? Explain.
b. What is the probability that it takes exactly 2 sessions to gain the patient's trust?
c. What is the probability that it takes at least 2 sessions to gain the patient's trust?

Michael Nartey
Michael Nartey
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01:18

Problem 14

MRA Company Projected Profits. The following table is a partial probability distribution for the MRA Company's projected profits ( $x=$ profit in $$\$ 1000 \mathrm{~s}$$ ) for the first year of operation (the negative value denotes a loss).

a. What is the proper value for $f(200)$ ? What is your interpretation of this value?
b. What is the probability that MRA will be profitable?
c. What is the probability that MRA will make at least $$\$ 100,000$$ ?

Nick Johnson
Nick Johnson
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01:08

Problem 15

The following table provides a probability distribution for the random variable x.

$$
\begin{array}{lr}
x & f(x) \\
3 & .25 \\
6 & 50 \\
9 & .25
\end{array}
$$

a. Compute $E(x)$, the expected value of $x$.
b. Compute $\sigma^2$, the variance of $x$.
c. Compute $\sigma$, the standard deviation of $x$.

Nick Johnson
Nick Johnson
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Problem 16

The following table provides a probability distribution for the random variable $y$.

$$
\begin{array}{rr}
y & f(y) \\
2 & 20 \\
4 & 30 \\
7 & 40 \\
8 & 10
\end{array}
$$

a. Compute $E(y)$.
b. Compute $\operatorname{Var}(y)$ and $\sigma$.

Rashmi Sinha
Rashmi Sinha
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02:41

Problem 17

Golf Scores. During the summer of 2018, Coldstream Country Club in Cincinnati, Ohio, collected data on 443 rounds of golf played from its white tees. The data for each golfer's score on the twelfth hole are contained in the DATAfile Coldstream12.
a. Construct an empirical discrete probability distribution for the player scores on the twelfth hole.
b. A par is the score that a good golfer is expected to get for the hole. For hole number 12, par is four. What is the probability of a player scoring less than or equal to par on hole number 12 ?
c. What is the expected score for hole number 12 ?
d. What is the variance for hole number 12 ?
c. What is the standard deviation for hole number 12 ?

Nick Johnson
Nick Johnson
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05:51

Problem 18

Water Supply Stoppages. The following data has been collected on the number of times that owner-occupied and renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months.

a. Define a random variable $x=$ number of times that owner-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let $x=4$ represent 4 or more times.)
b. Compute the expected value and variance for $x$.
c. Define a random variable $y=$ number of times that renter-occupied units had a water supply stoppage lasting 6 or more hours in the past 3 months and develop a probability distribution for the random variable. (Let $y=4$ represent 4 or more times.)
d. Compute the expected value and variance for $y$.
e. What observations can you make from a comparison of the number of water supply stoppages reported by owner-occupied units versus renter-occupied units?

Nick Johnson
Nick Johnson
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Problem 19

New Tax Accounting Clients. New legislation passed in 2017 by the U.S. Congress changed tax laws that affect how many people file their taxes in 2018 and beyond. These tax law changes will likely lead many people to seek tax advice from their accountants (The New York Times). Backen and Hayes LLC is an accounting firm in New York state. The accounting firm believes that it may have to hire additional accountants to assist with the increased demand in tax advice for the upcoming tax season. Backen and Hayes LLC has developed the following probability distribution for $x=$ number of new clients seeking tax advice.
a. Is this a valid probability distribution? Explain.
b. What is the probability that Backen and Hayes LLC will obtain 40 or more new clients?
c. What is the probability that Backen and Hayes LLC will obtain fewer than 35 new clients?
d. Compute the expected value, variance, and standard deviation of $x$.

Jason Gerber
Jason Gerber
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Problem 20

Automobile Insurance Damage Claims. The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance follows.

a. Use the expected collision payment to determine the collision insurance premium that would enable the company to break even.
b. The insurance company charges an annual rate of $$\$ 520$$ for the collision coverage. What is the expected value of the collision policy for a policyholder?

Rashmi Sinha
Rashmi Sinha
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03:07

Problem 21

IS Managers Job Satisfaction. The following probability distributions of job satisfaction scores for a sample of information systems (IS) senior executives and middle managers range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied).
a. What is the expected value of the job satisfaction score for senior executives?
b. What is the expected value of the job satisfaction score for middle managers?
c. Compute the variance of job satisfaction scores for executives and middle managers.
d. Compute the standard deviation of job satisfaction scores for both probability distributions.
e. Compare the overall job satisfaction of senior executives and middle managers.

Nick Johnson
Nick Johnson
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01:31

Problem 22

Carolina Industries Product Demand. The demand for a product of Carolina Industries varies greatly from month to month. The probability distribution in the following table, based on the past two years of data, shows the company's monthly demand.
a. If the company bases monthly orders on the expected value of the monthly demand, what should Carolina's monthly order quantity be for this product?
b. Assume that each unit demanded generates $$\$ 70$$ in revenue and that each unit ordered costs $$\$ 50$$. How much will the company gain or lose in a month if it places an order based on your answer to part (a) and the actual demand for the item is 300 units?

Nick Johnson
Nick Johnson
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Problem 23

Coffee Consumption. In Gallup's Annual Consumption Habits Poll, telephone interviews were conducted for a random sample of 1014 adults aged 18 and over. One of the questions was, "How many cups of coffee, if any, do you drink on an average day?" The following table shows the results obtained (Gallup website).

Define a random variable $x=$ number of cups of coffee consumed on an average day.
Let $x=4$ represent four or more cups.
a. Develop a probability distribution for $x$.
b. Compute the expected value of $x$.
c. Compute the variance of $x$.
d. Suppose we are only interested in adults who drink at least one cup of coffee on an average day. For this group, let $y=$ the number of cups of coffee consumed on an average day. Compute the expected value of $y$ and compare it to the expected value of $x$.

Jason Gerber
Jason Gerber
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04:17

Problem 24

Computer Company Plant Expansion. The J. R. Ryland Computer Company is considering a plant expansion to enable the company to begin production of a new computer product. The company's president must determine whether to make the expansion a medium- or large-scale project. Demand for the new product is uncertain, which for planning purposes may be low demand, medium demand, or high demand. The probability estimates for demand are $.20, .50$, and .30 , respectively. Letting $x$ and y indicate the annual profit in thousands of dollars, the firm's planners developed the following profit forecasts for the medium- and large-scale expansion projects.

a. Compute the expected value for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of maximizing the expected profit?
b. Compute the variance for the profit associated with the two expansion alternatives. Which decision is preferred for the objective of minimizing the risk or uncertainty?

Michael Nartey
Michael Nartey
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Problem 25

Given below is a bivariate distribution for the random variables $x$ and $y$.

$$
\begin{array}{ccc}
f(x, y) & x & y \\
2 & 50 & 80 \\
-5 & 30 & 50 \\
.3 & 40 & 60
\end{array}
$$

a. Compute the expected value and the variance for $x$ and $y$.
b. Develop a probability distribution for $x+y$.
c. Using the result of part (b), compute $E(x+y)$ and $\operatorname{Var}(x+y)$.
d. Compute the covariance and correlation for $x$ and $y$. Are $x$ and $y$ positively related, negatively related, or unrelated?
e. Is the variance of the sum of $x$ and $y$ bigger, smaller, or the same as the sum of the individual variances? Why?

Jason Gerber
Jason Gerber
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Problem 26

A person is interested in constructing a portfolio. Two stocks are being considered. Let $x=$ percent return for an investment in stock 1 , and $y=$ percent return for an investment in stock 2 . The expected return and variance for stock 1 are $E(x)=8.45 \%$ and $\operatorname{Var}(x)=25$. The expected retum and variance for stock $2 \operatorname{arc} E(y)=3.20 \%$ and $\operatorname{Var}(y)=1$. The covariance between the retums is $\sigma_{x y}=-3$.
a. What is the standard deviation for an investment in stock 1 and for an investment in stock 2 ? Using the standard deviation as a measure of risk, which of these stocks is the riskier investment?
b. What is the expected return and standard deviation, in dollars, for a person who invests $\$ 500$ in stock 1 ?
c. What is the expected percent return and standard deviation for a person who constructs a portfolio by investing $50 \%$ in each stock?
d. What is the expected percent return and standard deviation for a person who constructs a portfolio by investing $70 \%$ in stock 1 and $30 \%$ in stock 2 ?
e. Compute the correlation coefficient for $x$ and $y$ and comment on the relationship between the returns for the two stocks.

Jason Gerber
Jason Gerber
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02:04

Problem 27

Canadian Restaurant Ratings. The Chamber of Commerce in a Canadian city has conducted an evaluation of 300 restaurants in its metropolitan area. Each restaurant received a rating on a 3-point scale on typical meal price (1 least expensive to 3 most expensive) and quality ( 1 lowest quality to 3 greatest quality). A crosstabulation of the rating data is shown below. Forty-two of the restaurants received a rating of 1 on quality and 1 on meal price. 39 of the restaurants received a rating of 1 on quality and 2 on meal price, and so on. Forty-eight of the restaurants received the highest rating of 3 on both quality and meal price.a. Develop a bivariate probability distribution for quality and meal price of a randomly selected restaurant in this Canadian city. Let $x=$ quality rating and $y=$ meal price.
b. Compute the expected value and variance for quality rating, $x$.
c. Compute the expected value and variance for meal price, $y$.
d. The $\operatorname{Var}(x+y)=1.6691$. Compute the covariance of $x$ and $y$. What can you say about the relationship between quality and meal price? Is this what you would expect?
e. Compute the correlation coefficient between quality and meal price. What is the strength of the relationship? Do you suppose it is likely to find a low-cost restaurant in this city that is also high quality? Why or why not?

Dominador Tan
Dominador Tan
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Problem 28

Printer Manufacturing Costs. PortaCom has developed a design for a high-quality portable printer. The two key components of manufacturing cost are direct labor and parts. During a testing period, the company has developed prototypes and conducted extensive product tests with the new printer. PortaCom's engineers have developed the bivariate probability distribution shown below for the manufacturing costs. Parts cost (in dollars) per printer is represented by the random variable $x$ and direct labor cost (in dollars) per printer is represented by the random variable $y$. Management would like to use this probability distribution to estimate manufacturing costs.

a. Show the marginal distribution of direct labor cost and compute its expected value, variance, and standard deviation.
b. Show the marginal distribution of parts cost and compute its expected value, variance, and standard deviation.
c. Total manufacturing cost per unit is the sum of direct labor cost and parts cost. Show the probability distribution for total manufacturing cost per unit.
d. Compute the expected value, variance, and standard deviation of total manufacturing cost per unit.
c. Are direct labor and parts costs independent? Why or why not? If you conclude that they are not, what is the relationship between direct labor and parts cost?
f. PortaCom produced 1500 printers for its product introduction. The total manufacturing cost was $$\$ 198,350$$. Is that about what you would expect? If it is higher or lower, what do you think may have caused it?

Aishwarya Krishnakumar
Aishwarya Krishnakumar
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07:29

Problem 29

Investment Portfolio of Index Fund and Core Bonds Fund. J.P. Morgan Asset Management publishes information about financial investments. Between 2002 and 2011, the expected return for the S\&P 500 was $5.04 \%$ with a standard deviation of $19.45 \%$ and the expected return over that same period for a core bonds fund was $5.78 \%$ with a standard deviation of $2.13 \%$ (J.P. Morgan Asset Management, Gaide to the Markets). The publication also reported that the correlation between the S\&P 500 and core bonds is -32 . You are considering portfolio investments that are composed of an $S \& P 500$ index fund and a core bonds fund.
a. Using the information provided, determine the covariance between the S\&P 500 and core bonds.
b. Construct a portfolio that is $50 \%$ invested in an S\&P 500 index fund and $50 \%$ in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio?
c. Construct a portfolio that is $20 \%$ invested in an S\&P 500 index fund and $80 \%$ invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio?
d. Construct a portfolio that is $80 \%$ irvested in an $\$ \& P 500$ index fund and $20 \%$ invested in a core bonds fund. In percentage terms, what are the expected return and standard deviation for such a portfolio?
c. Which of the portfolios in parts (b), (c), and (d) has the largest expected return? Which has the smallest standard deviation? Which of these portfolios is the best investment?
f. Discuss the advantages and disadvantages of investing in the three portfolios in parts (b), (c), and (d). Would you prefer investing all your money in the S\&P 500 index, the core bonds fund, or one of the three portfolios? Why?

Dominador Tan
Dominador Tan
Numerade Educator
01:15

Problem 30

Investment Fund Including REITs. In addition to the information in exercise 29 on the S\&P 500 and core bonds, J.P. Morgan Asset Management reported that the expected return for real estate investment trusts (REITs) during the same time period was $13.07 \%$ with a standard deviation of $23.17 \%$ (J.P. Morgan Asser Management, Gaide to the Markets). The correlation between the S\&P 500 and REITs is .74 and the correlation between core bonds and REITs is -.04 . You are considering portfolio investments that are composed of an S\&P 500 index fund and REITs as well as portfolio investments composed of a core bonds fund and REITs.
a. Using the information provided here and in exercise 29 , determine the covariance between the S\&P 500 and REITs and between core bonds and REITs.
b. Construct a portfolio that is $50 \%$ invested in an S\&P 500 fund and $50 \%$ invested in REITs. In percentage terms, what are the expected return and standard deviation for such a portfolio?
c. Construct a portfolio that is $50 \%$ invested in a core bonds fund and $50 \%$ invested in REITs. In percentage terms, what are the expected return and standard deviation for such a portfolio?
d. Construct a portfolio that is $80 \%$ invested in a core bonds fund and $20 \%$ invested in REITs. In percentage terms, what are the expected return and standard deviation for such a portfolio?
c. Which of the portfolios in parts (b), (c), and (d) would you recommend to an aggressive investor? Which would you recommend to a conservative investor? Why?

Dominador Tan
Dominador Tan
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Problem 31

Consider a binomial experiment with two trials and $p=A$.
a. Draw a tree diagram for this experiment (see Figure 5.3).
b. Compute the probability of one success, $f(1)$.
c. Compute $f(0)$.
d. Compute $f(2)$.
e. Compute the probability of at least one success.
f. Compute the expected value, variance, and standard deviation.

Jason Gerber
Jason Gerber
Numerade Educator
03:59

Problem 33

Consider a binomial experiment with $n=20$ and $p=.70$.
a. Compute $f(12)$.
b. Compute $f(16)$.
c. Compute $P(x \geq 16)$.
d. Compute $P(x \leq 15)$.
e. Compute $E(x)$.
f. Compute $\operatorname{Var}(x)$ and $\sigma$.

Andrew Kim
Andrew Kim
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Problem 34

How Teenagers Listen to Music. For its Music 360 survey, Nielsen Co, asked teenagers and adults how each group has listened to music in the past 12 months. Nearly two-thirds of U.S. teenagers under the age of 18 say they use Google Inc.'s videosharing site to listen to music and $35 \%$ of the teenagers said they use Pandora Media Inc.'s custom online radio service (The Wall Street Journal). Suppose 10 teenagers are selected randomly to be interviewed about how they listen to music.
a. Is randomly selecting 10 teenagers and asking whether or not they use Pandora Media Inc.'s online service a binomial experiment?
b. What is the probability that none of the 10 teenagers uses Pandora Media Inc.'s online radio service?
c. What is the probability that 4 of the 10 teenagers use Pandora Media Inc.'s online radio service?
d. What is the probability that at least 2 of the 10 teenagers use Pandora Media Inc.'s online radio service?

Jason Gerber
Jason Gerber
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Problem 35

Appeals for Medicare Service. The Center for Medicare and Medical Services reported that there were 295,000 appeals for hospitalization and other Part A Medicare service. For this group, $40 \%$ of first-round appeals were successful (The Wall Street Journal). Suppose 10 first-round appeals have just been received by a Medicare appeals office.
a. Compute the probability that none of the appeals will be successful.
b. Compute the probability that exactly one of the appeals will be successful.
c. What is the probability that at least two of the appeals will be successful?
d. What is the probability that more than half of the appeals will be successful?

Jason Gerber
Jason Gerber
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01:29

Problem 36

Number of Defective Parts. When a new machine is functioning properly, only $3 \%$ of the items produced are defective. Assume that we will randomly select two parts produced on the machine and that we are interested in the number of defective parts found.
a. Describe the conditions under which this situation would be a binomial experiment
h. Draw a tree diagram similar to Figure 5.4 showing this problem as a two-trial experiment.
c. How many experimental outcomes result in exactly one defect being found?

Dominador Tan
Dominador Tan
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Problem 37

Americans Saving for Retirement. According to a 2018 survey by Bankrate.com, $20 \%$ of adults in the United States save nothing for retirement (CNBC website). Suppose that 15 adults in the United Statex are selected randomly.
a. Is the selection of the 15 adults a binomial experiment? Explain.
b. What is the probability that all of the selected adults save nothing for retirement?
c. What is the probability that exactly five of the selected adults save nothing for retirement?
d. What is the probability that at least one of the selected adults saves nothing for retirement?

James Kiss
James Kiss
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Problem 38

Detecting Missile Attacks. Military radar and missile detection systems are designed to warn a country of an enemy attack. A reliability question is whether a detection system will be able to identify an attack and issue a waming. Assume that a particular detection system has a . 90 probability of detecting a missile attack. Use the binomial probability distribution to answer the following questions.
a. What is the probability that a single detection system will detect an attack?
b. If two detection systems are installed in the same area and operate independently, what is the probability that at least one of the systems will detect the attack?
c. If three systems are installed, what is the probability that at least one of the systems will detect the attack?
d. Would you recommend that multiple detection systems be used? Explain.

Jason Gerber
Jason Gerber
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Problem 39

. Web Browser Market Share. Market-share-analysis company Net Applications monitors and reports on Internet browser usage. According to Net Applications, in the summer of 2014 , Google's Chrome browser exceeded a $20 \%$ market share for the first time, with a $20.37 \%$ share of the browser market (Forbes website). For a randomly selected group of 20 Intemet browser users, answer the following questions.
a. Compute the probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser.
b. Compute the probability that at least 3 of the 20 Internet browser users use Chrome as their Internet browser.
c. For the sample of 20 Internet browser users, compute the expected number of Chrome users.
d. For the sample of 20 Internet browser users, compute the variance and standard deviation for the number of Chrome users.

Jason Gerber
Jason Gerber
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Problem 40

Contributing to Household Income. Suppose that a random sample of fifteen 18- to 34 -year-olds living with their parents is selected and asked if they contribute to household expenses.
a. Is the selection of the fifteen 18 - to 34 -year-olds living with their parents a binomial experiment? Explain.
b. If the sample shows that none of the fifteen 18 - to 34 -year-olds living with their parents contribute to household expenses, would you question the results of the Pew Research Study? Explain.
c. What is the probability that at least 10 of the fifteen 18 - to 34 -year-olds living with their parents contribute to household expenses?

Jason Gerber
Jason Gerber
Numerade Educator
05:02

Problem 41

Introductory Statistics Course Withdrawals. A university found that $20 \%$ of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course.
a. Compute the probability that 2 or fewer will withdraw.
b. Compute the probability that exactly 4 will withdraw.
c. Compute the probability that more than 3 will withdraw.
d Comnute the exnected number of withdrawals

Sheryl Ezze
Sheryl Ezze
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Problem 42

Consider a binomial experiment with $n=10$ and $p=.10$.
a. Compute $f(0)$.
b. Compute $f(2)$.
c. Compute $P(x \leq 2)$.
d. Compute $P(x \geq 1)$.
c. Compute $E(x)$.
f. Compute $\operatorname{Var}(x)$ and $\sigma$.

Shu Naito
Shu Naito
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Problem 42

State of the Nation Survey. Suppose a sample of 20 Americans is selected as part of a study of the state of the nation. The Americans in the sample are asked whether or not they are satisfied with the way things are going in the United States.
State of the Nation Survey. Suppose a sample of 20 Americans is selected as part of a study of the state of the nation. The Americans in the sample are asked whether or not they are satisfied with the way things are going in the United States.

a. Compute the probability that exactly 4 of the 20 Americans surveyed are satisfied with the way things are going in the United States.
b. Compute the probability that at least 2 of the Americans surveyed are satisfied with the way things are going in the United States.
c. For the sample of 20 Americans, compute the expected number of Americans who are satisfied with the way things are going in the United States.
d. For the sample of 20 Americans, compute the variance and standard deviation of the number of Americans who are satisfied with the way things are going in the United States.

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 43

Tracked Emails. According to a 2017 Wired magazine article, $40 \%$ of emails that are received are tracked using software that can tell the email sender when, where, and on what type of device the email was opened (Wired magazine website). Suppose we randomly select 50 received emails.
a. What is the expected number of these emails that are tracked?
b. What are the variance and standard deviation for the number of these emails that are tracked?

Jason Gerber
Jason Gerber
Numerade Educator
02:14

Problem 44

Consider a Poisson distribution with $\mu=3$.
a. Write the appropriate Poisson probability function.
b. Compute $f(2)$.
c. Compute $f(1)$.
d. Compute $P(x \geq 2)$.

Foster Wisusik
Foster Wisusik
Numerade Educator
02:22

Problem 45

Consider a Poisson distribution with a mean of two occurrences per time period.
a. Write the appropriate Poisson probability function.
b. What is the expected number of occurrences in three time periods?
c. Write the appropriate Poisson probability function to determine the probability of $x$ occurrences in three time periods.
d. Compute the probability of two occurrences in one time period.
c. Compute the probability of six occurrences in three time periods.
f. Compute the probability of five occurrences in two time periods.

Nick Johnson
Nick Johnson
Numerade Educator
02:35

Problem 46

Regional Airways Calls. Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways.
a. Compute the probability of receiving three calls in a 5 -minute interval of time.
b. Compute the probability of receiving exactly 10 calls in 15 minutes.
c. Suppose no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time?
What is the probability that none will be waiting?
d. If no calls are currently being processed, what is the probability that the agent can take 3 minutes for personal time without being interrupted by a call?

Nick Johnson
Nick Johnson
Numerade Educator
02:42

Problem 47

911 Calls. Emergency 911 calls to a small municipality in Idaho come in at the rate of one every 2 minutes.
a. What is the expected number of 911 calls in one hour?
b. What is the probability of theee 911 calls in five minutes?
c. What is the probability of no 911 calls in a five-minute period?

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 48

Motor Vehicle Accidents in New York City. In a one-year period, New York City had a total of 11,232 motor vehicle accidents that occurred on Monday through Friday between the hours of 3 P.M. and 6 P.M. (New York State Department of Motor Vehicles website). This corresponds to mean of 14.4 accidents per hour.
a. Compute the probability of no accidents in a 15 -minute period.
b. Compute the probability of at least one accident in a 15 -minute period.
c. Compute the probability of four or more accidents in a 15 -minute period.

Jason Gerber
Jason Gerber
Numerade Educator
02:25

Problem 49

Airport Passenger-Screening Facility. Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 10 passengers per minute.
a. Compute the probability of no arrivals in a one-minute period.
b. Compute the probability that three or fewer passengers arrive in a one-minute period.
c. Compute the probability of no arrivals in a 15 -second period.
d. Compute the probability of at least one arrival in a 15 -second period.

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 50

Tornadoes in Colorado. According to the National Occanic and Atmospheric Administration (NOAA), the state of Colorado averages 18 tomadoes every June (NOAA website). (Note: There are 30 days in June.)
a. Compute the mean number of tornadoes per day.
b. Compute the probability of no tornadoes during a day.
c. Compute the probability of exactly one tornado during a day.
d. Compute the probability of more than one tornado during a day.

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 51

Emails Received. According to a 2017 survey conducted by the technology market research firm The Radicati Group, U.S. office workers receive an average of 121 emails per day (Entrepreneur magazine website). Assume the number of emails received per hour follows a Poisson distribution and that the average number of emails received per hour is five.
a. What is the probability of receiving no emails during an hour?
b. What is the probability of receiving at least three emails during an hour?
c. What is the expected number of emails received during 15 minutes?
d. What is the probability that no emails are received during 15 minutes?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 53

Emails Received. According to a 2017 survey conducted by the technology market research firm The Radicati Group, U.S. office workers receive an average of 121 emails per day (Entrepreneur magazine website). Assume the number of emails received per hour follows a Poisson distribution and that the average number of emails received per hour is five.
a. What is the probability of receiving no emails during an hour?
b. What is the probability of receiving at least three emails during an hour?
c. What is the expected number of emails received during 15 minutes?
d. What is the probability that no emails are received during 15 minutes?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 54

Online Holiday Shopping. More and more shoppers prefer to do their holiday shopping online from companies such as Amazon. Suppose we have a group of 10 shoppers; 7 prefer to do their holiday shopping online and 3 prefer to do their holiday shopping in stores. A random sample of 3 of these 10 shoppers is selected for a more in-depth study of how the economy has impacted their shopping behavior.
a. What is the probability that exactly 2 prefer shopping online?
b. What is the probability that the majority (either 2 or 3 ) prefer shopping online?

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 55

Playing Blackjack. Blackjack, or twenty-one as it is frequently called, is a popular gambling game played in casinos. A player is dealt two cards. Face cards (jacks, queens, and kings) and tens have a point value of 10 . Aces have a point value of 1 or 11. A 52 -card deck contains 16 cards with a point value of 10 (jacks, queens, kings, and tens) and four aces.
a. What is the probability that both cards dealt are aces or 10 -point cards?
b. What is the probability that both of the cards are aces?
c. What is the probability that both of the cards have a point value of 10 ?
d. A blackjack is a 10 -point card and an ace for a value of 21 . Use your answers to parts (a), (b), and (c) to determine the probability that a player is dealt blackjack. (Hint: Part (d) is not a hypergeometric problem. Develop your own logical relationship as to how the hypergeometric probabilities from parts (a), (b), and (c) can be combined to answer this question.)

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 56

Computer Company Benefits Questionnaire. Axline Computers manufactures personal computers at two plants, one in Texas and the other in Hawaii. The Texas plant has 40 employees; the Hawaii plant has 20 . A random sample of 10 employees is to be asked to fill out a benefits questionnaire.
a. What is the probability that none of the employees in the sample works at the plant in Hawaii?
b. What is the probability that one of the employees in the sample works at the plant in Hawaii?
c. What is the probability that two or more of the employees in the sample work at the plant in Hawaii?
d. What is the probobility that 9 of the employees in the sample work at the plant in Texas?

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 57

Business Meal Reimbursement. The Zagat Restaurant Survey provides food, decor, and service ratings for some of the top restaurants across the United States. For 15 restaurants located in Boston, the average price of a dinner, including one drink and tip. was $$\$ 48.60$$. You are leaving on a business trip to Boston and will eat dinner at three of these restaurants. Your company will reimburse you for a maximum of $$\$ 50$$ per dinner. Business associates familiar with these restaurants have told you that the meal cost at one-third of these restaurants will exceed $$\$ 50$$. Suppose that you randomly select three of these restaurants for dinner.
a. What is the probability that none of the meals will exceed the cost covered by your company?
b. What is the probability that one of the meals will exceed the cost covered by your company?
c. What is the probability that two of the meals will exceed the cost covered by your company?
d. What is the probability that all three of the meals will exceed the cost covered by your company?

William Fairburn
William Fairburn
Numerade Educator
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Problem 58

TARP Funds. The Troubled Asset Relief Program (TARP). passed by the U.S. Congress in October 2008, provided $$\$ 700$$ billion in assistance for the struggling U.S. economy. Over $$\$ 200$$ billion was given to troubled financial institutions with the hope that there would be an increase in lending to help jump-start the economy. But three months later, a Federal Reserve survey found that two-thirds of the banks that had received TARP funds had tightened terms for business loans (The Wall Street Journal').

For the purposes of this exercise, assume that you will randomly select 3 of these 10 banks for a study that will continue to monitor hank lending practices. Let $x$ be a random variable indicating the number of banks in the study that had increased lending.
a. What is $f(0)$ ? What is your interpretation of this value?
b. What is $f(3)$ ? What is your interpretation of this value?
c. Compute $f(1)$ and $f(2)$. Show the probability distribution for the number of banks in the study that had increased lending. What value of $x$ has the highest probability?
d. What is the probability that the study will have at least one bank that had increased lending?
e. Compute the expected value, variance, and standard deviation for the random variable.

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 59

Wind Conditions and Boating Accidents. The U.S. Coast Guard (USCG) provides a wide variety of information on boating accidents including the wind condition at the time of the accident. The following table shows the results obtained for 4401 accidents

$$
\begin{array}{lc}
\text { Wind Condition } & \begin{array}{c}
\text { Percentage of } \\
\text { Accidents }
\end{array} \\
\text { None } & 9.6 \\
\text { Light } & 57.0 \\
\text { Moderate } & 23.8 \\
\text { Strong } & 7.7 \\
\text { Storm } & 1.9
\end{array}
$$

Let $x$ be a random variable reflecting the known wind condition at the time of each accident. Set $x=0$ for none, $x=1$ for light, $x=2$ for moderate, $x=3$ for strong, and $x=4$ for storm.
a. Develop a probability distribution for $x$.
b. Compute the expected value of $x$.
c. Compute the variance and standard deviation for $x$.
d. Comment on what your results imply about the wind conditions during boating accidents.

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 60

Wait Times at Car Repair Garages. The Car Repair Ratings website provides consumer reviews and ratings for garages in the United States and Canada. The time customers wait for service to be completed is one of the categories rated. The following table provides a summary of the wait-time ratings ( $1=$ Slow/Delays; $10=$ Quick/On Time) for 40 randomly selected garages located in the province of Ontario, Canada.

a. Develop a probability distribution for $x=$ wait-time rating.
b. Any garage that receives a wait-time rating of at least 9 is considered to provide outstanding service. If a consumer randomly selects one of the 40 garages for their next car service, what is the probability the garage selected will provide outstanding waittime service?
c. What is the expected value and variance for $x$ ?
d. Suppose that 7 of the 40 garages reviewed were new car dealerships. Of the 7 new car dealerships, two were rated as providing outstanding wait-time service. Compare the likelihood of a new car dealership achieving an outstanding waittime service rating as compared to other types of service providers.

Jason Gerber
Jason Gerber
Numerade Educator
02:05

Problem 61

Expense Forecasts. The budgeting process for a midwestern college resulted in expense forecasts for the coming year (in $$\$$ millions) of $\$ 9, \$ 10, \$ 11, \$ 12$$, and $$\$ 13$$. Because the actual expenses are unknown, the following respective probabilities are assigned: $3, .2, .25, .05$, and .2 .
a. Show the probability distribution for the expense forecast.
b. What is the expected value of the expense forecast for the coming year?
c. What is the variance of the expense forecast for the coming year?
d. If income projections for the year are estimated at $$\$ 12$$ million, comment on the financial position of the college.

Nick Johnson
Nick Johnson
Numerade Educator
02:30

Problem 62

Bookstore Customer Purchases. A bookstore at the Hartsfield-Jackson Airport in Atlanta sells reading materials (paper-back books, newspapers, magazines) as well as snacks (peanuts, pretzels, candy, etc.). A point-of-sale terminal collects a variety of information about customer purchases. Shown below is a table showing the number of snack items and the number of items of reading material purchased by the most recent 600 customers.

a. Using the data in the table construct an empirical discrete bivariate probability distribution for $x=$ number of snack items and $y=$ number of reading materials in a randomly selected customer purchase. What is the probability of a customer purchase consisting of one item of reading materials and two snack items? What is the probability of a customer purchasing one snack item only? Why is the probability $f(x=0, y=0)=0$ ?
b. Show the marginal probability distribution for the number of snack items purchased. Compute the expected value and variance.
c. What is the expected value and variance for the number of reading materials purchased by a customer?
d. Show the probability distribution for $t=$ total number of items in a customer purchase. Compute its expected value and variance.
c. Compute the covariance and correlation coefficient between $x$ and $y$. What is the relationship, if any, between the number of reading materials and number of snacks purchased on a customer visit?

Dominador Tan
Dominador Tan
Numerade Educator
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Problem 63

Creating a Diversified Investment Portfolio. The Knowles/Armitage (KA) group at Merrill Lynch advises clients on how to create a diversified investment portfolio. One of the investment alternatives they make available to clients is the All World Fund composed of global stocks with good dividend yields. One of their clients is interested in a portfolio consisting of imvestment in the All World Fund and a treasury bond fund. The expected percent return of an investment in the All World Fund is $7.80 \%$ with a standard deviation of $18.90 \%$. The expected percent return of an investment in a treasury bond fund is $5.50 \%$ and the standard deviation is $4.60 \%$. The covariance of an investment in the All World Fund with an investment in a treasury bond fund is -12.4 .
a. Which of the funds would be considered the more risky? Why?
b. If KA recommends that the client invest $75 \%$ in the All World Fund and $25 \%$ in the treasury bond fund, what is the expected percent return and standard deviation for such a portfolio? What would be the expected return and standard deviation, in dollars, for a client investing $$\$ 10,000$$ in such a portfolio?
c. If KA recommends that the client invest $25 \%$ in the All World Fund and $75 \%$ in the treasury bond fund, what is the expected return and standard deviation for such a portfolio? What would be the expected return and standard deviation, in dollars, for a client investing $$\$ 10,000$$ in such a portfolio?
d. Which of the portfolios in parts (b) and (c) would you recommend for an aggressive investor? Which would you recommend for a conservative investor? Why?

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 64

Giving up Technology. A Pew Research Survey asked adults in the United States which technologies would be "very hard" to give up. The following responses were obtained: Internet $53 \%$, smartphone $49 \%$, email $36 \%$, and land-line phone $28 \%$ (USA Today website).
a. If 20 adult Internet users are surveyed, what is the probability that 3 users will report that it would be very hard to give it up?
b. If 20 adults who own a land-line phone are surveyed, what is the probability that 5 or fewer will report that it would be very hard to give it up?
c. If 2000 owners of smartphones were surveyed, what is the expected number that will report that it would be very hard to give it up?
d. If 2000 users of email were surveyed, what is expected number that will report that it would be very hard to give it up? What is the variance and standard deviation?

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 65

Investing in the Stock Market. According to a 2017 Gallup survey, the percentage of individuals in the United States who are invested in the stock market by age is as shown in the following table (Gallup website).

Suppose Gallup wishes to complete a follow-up survey to find out more about the specific type of stocks people in the United States are purchasing.
a. How many 18 to 29 year olds must be sampled to find at least 50 who invest in the stock market?
b. How many people 65 years of age and older must be sampled to find at least 50 who invest in the stock market?
c. If 1000 individuals are randomly sampled, what is the expected number of 18 to 29 year olds who invest in the stock market in this sample? What is the standard deviation of the number of 18 to 29 year olds who invest in the stock market?
d. If 1000 individuals are randomly sampled, what is the expected number of those 65 and older who invest in the stock market in this sample? What is the standard deviation of the number of those 65 years of age and older who invest in the stock market?

Jason Gerber
Jason Gerber
Numerade Educator
04:05

Problem 66

Acceptance Sampling. Many companies use a quality control technique called acceptance sampling to monitor incoming shipments of parts, raw materials, and so on. In the electronics industry, component parts are commonly shipped from suppliers in large lots. Inspection of a sample of $n$ components can be viewed as the $n$ trials of a binomial experiment. The outcome for each component tested (trial) will be that the component is classified as good or defective. Reynolds Electronics accepts a lot from a particular supplier if the defective components in the lot do not exceed $1 \%$. Suppose a random sample of five items from a recent shipment is texted.
a. Assume that $1 \%$ of the shipment is defective. Compute the probability that no items in the sample are defective.
b. Assume that $1 \%$ of the shipment is defective. Compute the probability that exactly one item in the sample is defective.
c. What is the probability of observing one or more defective items in the sample if $1 \%$ of the shipment is defective?
d. Would you feel comfortable accepting the shipment if one item was found to be defective? Why or why not?

Foster Wisusik
Foster Wisusik
Numerade Educator
01:47

Problem 67

Americans with at Least a Two-Year Degree. PBS News Hour reported in 2014 that $39.4 \%$ of Americans between the ages of 25 and 64 have at least a two-year college degree (PBS website). Assume that 50 Americans between the ages of 25 and 64 are selected randomly.
a. What is the expected number of people with at least a two-year college degree?
b. What are the variance and standard deviation for the number of people with at least a two-year college degree?

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 68

Choosing a Home Builder. Mahoncy Custom Home Builders, Inc. of Canyon Lake, Texas, asked visitors to their website what is most important when choosing a home builder. Possible responses were quality, price, customer referral, years in business. and special features. Results showed that $23.5 \%$ of the respondents chose price as the most important factor (Mahoney Custom Homes website). Suppose a sample of 200 potential home buyers in the Canyon Lake area are selected.
a. How many people would you expect to choose price as the most important factor when choosing a home builder?
b. What is the standard deviation of the number of respondents who would choose price as the most important factor in selecting a home builder?
c. What is the standard deviation of the number of respondents who do not list price as the most important factor in selecting a home builder?

Jason Gerber
Jason Gerber
Numerade Educator
03:14

Problem 69

Arrivals to a Car Wash. Cars arrive at a car wash randomly and independently; the probability of an arrival is the same for any two time intervals of equal length. The mean arrival rate is 15 cars per hour. What is the probability that 20 or more cars will arrive during any given hour of operation?

Andrew Kim
Andrew Kim
Numerade Educator
01:01

Problem 70

Production Process Breakdowns. A new automated production process averages 1.5 breakdowns per day. Because of the cost associated with a breakdown, management is concerned about the possibility of having three or more breakdowns during a day. Assume that breakdowns occur randomly, that the probability of a breakdown is the same for any two time intervals of equal length, and that breakdowns in one period are independent of breakdowns in other periods. What is the probability of having three of more breakdowns during a day?

Foster Wisusik
Foster Wisusik
Numerade Educator
01:21

Problem 71

Small Business Failures. A regional director responsible for business development in the state of Pennsylvania is concerned about the number of small business failures. If the mean number of small business failures per month is 10 , what is the probability that exactly 4 small businesses will fail during a given month? Assume that the probability of a failure is the same for any two months and that the occurrence or nonoccurrence of a failure in any month is independent of failures in any other month.

Andrew Kim
Andrew Kim
Numerade Educator
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Problem 72

Bank Customer Arrivals. Customer arrivals at a bank are random and independent; the probability of an arrival in any one-minute period is the same as the probability of an arrival in any other one-minute period. Answer the following questions, assuming a mean arrival rate of three customers per minute.
a. What is the probability of exactly three arrivals in a one-minute period?
b. What is the probability of at least three arrivals in a one-minute period?

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 73

Poker Hands. A deck of playing cards contains 52 cards, four of which are aces. What is the probability that the deal of a five-card poker hand provides
a. A pair of aces?
b. Exactly one ace?
c. No aces?
d. At least one ace?

Jason Gerber
Jason Gerber
Numerade Educator
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Problem 74

Business School Student GPAs. According to U.S. News \& World Reparts, 7 of the top 10 graduate schools of business have students with an average undergraduate grade point average (GPA) of 3.50 or higher. Suppose that we randomly select 2 of the top 10 graduate schools of business.

a. What is the probability that exactly one school has students with an average undergraduate GPA of 3.50 or higher?
b. What is the probability that both schools have students with an average undergraduate GPA of 3.50 or higher?
c. What is the probability that neither sthool has students with an average undergraduate GPA of 3.50 or higher?

Jason Gerber
Jason Gerber
Numerade Educator