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Business Analytics

Jeffrey D. Camm, James J. Cochran, Michael J. Fry, Jeffrey W. Ohlmann

Chapter 5

Probability An Introduction to Modeling Uncertainty - all with Video Answers

Educators

Chapter Questions

02:25

Problem 1

TikTok User Ages. It is estimated that nearly half of all users of the social networking platform TikTok in the United States were between the ages of 10 and 29 in 2020 (Statista.com website). Suppose that the results below were received from a survey sent to a random sample of people in the United States regarding their use of TikTok.
$$
\begin{array}{ccc}
\text { Age Range of Respondent } & \text { Uses TikTok } & \text { Does Not Use TikTok } \\
10-19 & 36 & 44 \\
20-29 & 87 & 52 \\
30-39 & 42 & 89 \\
40+ & 44 & 138 \\
\hline \text { Total: } & 209 & 323
\end{array}
$$
a. Use the sample data above to compute the probability that a respondent to this survey uses TikTok.
b. What is the probability of a respondent to this survey in each age range (10-19, $20-29,30-39,40+$ ) using TikTok? Which age range has the highest probability of a respondent being a user of TikTok?

Trinity Steen
Trinity Steen
Numerade Educator
02:05

Problem 2

On-time arrivals, lost baggage, and customer complaints are three measures that are typically used to measure the quality of service being offered by airlines. Suppose that the following values represent the on-time arrival percentage, amount of lost baggage, and customer complaints for 10 U.S. airlines.
$$
\begin{array}{lccc}
& \begin{array}{c}
\text { On-Time } \\
\text { Arrivals (\%) }
\end{array} & \begin{array}{c}
\text { Mishandled Baggage } \\
\text { per 1,000 Passengers }
\end{array} & \begin{array}{c}
\text { Customer } \\
\text { Complaints per } \\
\text { 1,000 Passengers }
\end{array} \\
\text { Airline } & 83.5 & 0.87 & 1.50 \\
\text { Virgin America } & 79.1 & 1.88 & 0.79 \\
\text { JetBlue } & 87.1 & 1.58 & 0.91 \\
\text { AirTran Airways } & 86.5 & 2.10 & 0.73 \\
\text { Delta Air Lines } & 87.5 & 2.93 & 0.51 \\
\text { Alaska Airlines } & 77.9 & 2.22 & 1.05 \\
\text { Frontier Airlines } & 83.1 & 3.08 & 0.25 \\
\text { Southwest Airlines } & 85.9 & 2.14 & 1.74 \\
\text { US Airways } & 76.9 & 2.92 & 1.80 \\
\text { American Airlines } & 77.4 & 3.87 & 4.24 \\
\text { United Airlines } & & &
\end{array}
$$
a. Based on the data above, if you randomly choose a Delta Air Lines flight, what is the probability that this individual flight will have an on-time arrival?
b. If you randomly choose 1 of the 10 airlines for a follow-up study on airline quality ratings, what is the probability that you will choose an airline with less than two mishandled baggage reports per 1,000 passengers?
c. If you randomly choose 1 of the 10 airlines for a follow-up study on airline quality ratings, what is the probability that you will choose an airline with more than one customer complaint per 1,000 passengers?
d. What is the probability that a randomly selected AirTran Airways flight will not arrive on time?

Nick Johnson
Nick Johnson
Numerade Educator
07:05

Problem 3

Consider the random experiment of rolling a pair of six-sided dice. Suppose that we are interested in the sum of the face values showing on the dice.
a. How many outcomes are possible?
b. List the outcomes.
c. What is the probability of obtaining a value of 7 ?
d. What is the probability of obtaining a value of 9 or greater?

Pratyush Raitan
Pratyush Raitan
Numerade Educator
04:08

Problem 4

Only 15 college basketball programs have won more than a single NCAA Basketball Championship. Those 15 colleges are shown below along with their current NCAA Conference Affiliation and the number of NCAA Basketball Championships they have won. Suppose that the winner of one of these 60 NCAA Basketball Championships is selected at random.
$$
\begin{array}{llc}
\text { College } & \begin{array}{l}
\text { Current NCAA Conference } \\
\text { Affiliation }
\end{array} & \begin{array}{l}
\text { Number of NCAA Basketball } \\
\text { Championships }
\end{array} \\
\text { UCLA } & \text { Pac-12 } & 11 \\
\text { Kentucky } & \text { SEC } & 8 \\
\text { North Carolina } & \text { ACC } & 6 \\
\text { Duke } & \text { ACC } & 5 \\
\text { Indiana } & \text { Big Ten } & 5 \\
\text { Kansas } & \text { Big 12 } & 4 \\
\text { UConn } & \text { Big East } & 4 \\
\text { Villanova } & \text { Big East } & 3 \\
\text { Cincinnati } & \text { American } & 2 \\
\text { Florida } & \text { SEC } & 2 \\
\text { Louisville } & \text { ACC } & 2 \\
\text { Michigan State } & \text { Big Ten } & 2 \\
\text { NC State } & \text { ACC } & 2 \\
\text { Oklahoma State } & \text { Big 12 } & 2 \\
\text { San Francisco } & \text { West Coast } & 2 \\
\hline
\end{array}
$$
a. What is the probability that the selected winner is from UCLA?
b. What is the probability that the selected winner is from the Big Ten Conference?
c. What is the probability that the selected winner is from the ACC?
d. What is the probability that the selected winners is from either the Big Ten or the ACC?

Pratyush Raitan
Pratyush Raitan
Numerade Educator
03:00

Problem 5

Suppose that for a recent admissions class, an Ivy League college received 2,851 applications for early admission. Of this group, it admitted 1,033 students early, rejected 854 outright, and deferred 964 to the regular admission pool for further consideration. In the past, this school has admitted $18 \%$ of the deferred early admission applicants during the regular admission process. Counting the students admitted early and the students admitted during the regular admission process, the total class size was 2,375 . Let $E, R$, and $D$ represent the events that a student who applies for early admission is admitted early, rejected outright, or deferred to the regular admissions pool. LO 1, 2
a. Use the data to estimate $P(E), P(R)$, and $P(D)$.
b. Are events $E$ and $D$ mutually exclusive? Find $P(E \cap D)$.
c. For the 2,375 students who were admitted, what is the probability that a randomly selected student was accepted during early admission?
d. Suppose a student applies for early admission. What is the probability that the student will be admitted for early admission or be deferred and later admitted during the regular admission process?

Bailey Brooks
Bailey Brooks
Numerade Educator
02:57

Problem 6

Suppose that we have two events, $A$ and $B$, with $P(A)=0.50$, $P(B)=0.60$, and $P(A \cap B)=0.40$. LO 2, 3, 4
a. Find $P(A \mid B)$.
b. Find $P(B \mid A)$.
c. Are $A$ and $B$ independent? Why or why not?

Sandra Kudolo
Sandra Kudolo
Numerade Educator
04:01

Problem 7

Students taking the Graduate Management Admissions Test (GMAT) were asked about their undergraduate major and intent to pursue their MBA as a full-time or part-time student. A summary of their responses is shown in the table that follows.
$$
\begin{array}{lr|ccc|c}
& & \text { Business } & \text { Engineering } & \text { Other } & \text { Totals } \\
{ 2 - 6 } \text { Intended } & \text { Full-Time } & 352 & 197 & 251 & 800 \\
\text { Enrollment } & \text { Part-Time } & 150 & 161 & 194 & 505 \\
{ 2 - 5 } \text { Status } & \text { Totals } & 502 & 358 & 445 & 1,305
\end{array}
$$
a. Develop a joint probability table for these data.
b. Use the marginal probabilities of undergraduate major (business, engineering, or other) to comment on which undergraduate major produces the most potential MBA students.
c. If a student intends to attend classes full time in pursuit of an MBA degree, what is the probability that the student was an undergraduate engineering major?
d. If a student was an undergraduate business major, what is the probability that the student intends to attend classes full time in pursuit of an MBA degree?
e. Let $F$ denote the event that the student intends to attend classes full time in pursuit of an MBA degree, and let $B$ denote the event that the student was an undergraduate business major. Are events $F$ and $B$ independent? Justify your answer.

Nick Johnson
Nick Johnson
Numerade Educator
02:53

Problem 8

More than 40 million Americans are estimated to have at least one outstanding student loan to help pay college expenses (CNNMoney web site). Not all of these graduates pay back their debt in satisfactory fashion. Suppose that the following joint probability table shows the probabilities of student loan status and whether or not the student had received a college degree.
$$
\begin{aligned}
&\text { College Degree }\\
&\begin{array}{ll|ll|l}
& & \text { Yes } & \text { No } & \\
{ 2 - 5 } \begin{array}{l}
\text { Loan } \\
\text { Status }
\end{array} & \text { Satisfactory } & 0.26 & 0.24 & 0.50 \\
& \text { Delinquent } & 0.16 & 0.34 & 0.50 \\
& & 0.42 & 0.58 &
\end{array}
\end{aligned}
$$
a. What is the probability that a student with a student loan had received a college degree?
b. What is the probability that a student with a student loan had not received a college degree?
c. Given that the student has received a college degree, what is the probability that the student has a delinquent loan?
d. Given that the student has not received a college degree, what is the probability that the student has a delinquent loan?
e. What is the impact of dropping out of college without a degree for students who have a student loan?

Nick Johnson
Nick Johnson
Numerade Educator

Problem 9

The Human Resources Manager for Optilytics LLC is evaluating applications for the position of Senior Data Scientist. The file optilytics presents summary data of the applicants for the position. The collected data include the highest degree achieved and the years of work experience for each candidate as well as whether or not the candidate has experience using Alteryx, a software used for analytics. LO 2, 3, 5
a. Use a PivotTable in Excel to create a joint probability table showing the probabilities associated with a randomly selected applicant's use of Alteryx and highest degree achieved. Use this joint probability table to answer the questions below.
b. What are the marginal probabilities? What do they tell you about the probabilities associated with the use of Alteryx among the applicants and highest degree completed by applicants?
c. If the applicant has not used Alteryx, what is the probability that the highest degree completed by the applicant is a $\mathrm{PhD}$ ?
d. If the highest degree completed by the applicant is a bachelor's degree, what is the probability that the applicant has used Alteryx?
e. What is the probability that a randomly selected applicant has used Alteryx and whose highest completed degree is a $\mathrm{PhD}$ ?

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03:29

Problem 10

The U.S. Census Bureau is a leading source of quantitative data related to the people and economy of the United States. The crosstabulation below represents the number of households (thousands) and the household income by the highest level of education for the head of household (U.S. Census Bureau web site). Use this crosstabulation to answer the following questions.
$$
\begin{array}{l|rccc|r}
\text { Highest Level of } & \begin{array}{c}
\text { Under } \\
\text { Education }
\end{array} & \$ 25,000- & \$ 50,000- & \$ 100,000 \\
\text { High school graduate } & 9,880 & 9,970 & 9,441 & 3,482 & 32,773 \\
\hline \text { Bachelor's degree } & 2,484 & 4,164 & 7,666 & 7,817 & 22,131 \\
\text { Master's degree } & 685 & 1,205 & 3,019 & 4,094 & 9,003 \\
\text { Doctoral degree } & 79 & 160 & 422 & 1,076 & 1,737 \\
\hline \text { Total } & 13,128 & 15,499 & 20,548 & 16,469 & 65,644
\end{array}
$$
a. Develop a joint probability table.
b. What is the probability the head of one of these households has a master's degree or higher education?
c. What is the probability a household is headed by someone with a high school diploma earning $$\$ 100,000$$ or more?
d. What is the probability one of these households has an income below $$\$ 25,000$$ ?
e. What is the probability a household is headed by someone with a bachelor's degree earning less than $$\$ 25,000$$ ?
f. Are household income and educational level independent?

Nick Johnson
Nick Johnson
Numerade Educator
05:02

Problem 11

Cooper Realty is a small real estate company located in Albany, New York, that specializes primarily in residential listings. The company recently became interested in determining the likelihood of one of its listings being sold within a certain number of days. An analysis of company sales of 800 homes in previous years produced the following data.
a. If $A$ is defined as the event that a home is listed for more than 90 days before being sold, estimate the probability of $A$.
b. If $B$ is defined as the event that the initial asking price is under $$\$ 150,000$$, estimate the probability of $B$.
c. What is the probability of $A \cap B$ ?
d. Assuming that a contract was just signed to list a home with an initial asking price of less than $$\$ 150,000$$, what is the probability that the home will take Cooper Realty more than 90 days to sell?
e. Are events $A$ and $B$ independent?
$$
\begin{aligned}
&\text { Days Listed Until Sold }\\
&\begin{array}{ll|crr|c}
& & \text { Under } \mathbf{3 0} & \mathbf{3 1 - 9 0} & \text { Over } \mathbf{9 0} & \text { Total } \\
{ 2 - 6 } \text { Initial Asking Price } & \text { Under } \mathbf{\$ 1 5 0 , 0 0 0} & 50 & 40 & 10 & 100 \\
& \$ 150,000-\mathbf{\$ 1 9 9 , 9 9 9} & 20 & 150 & 80 & 250 \\
& \$ \mathbf{2 0 0 , 0 0 0} \mathbf{- \$ 2 5 0 , 0 0 0} & 20 & 280 & 100 & 400 \\
& \text { Over } \mathbf{\$ 2 5 0 , 0 0 0} & 10 & 30 & 10 & 50 \\
\text { Total } & 100 & 500 & 200 & 800
\end{array}
\end{aligned}
$$

Georgiann Andersen
Georgiann Andersen
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Problem 12

Computing Probabilities. The prior probabilities for events $A_1$ and $A_2$ are $P\left(A_1\right)=0.40$ and $P\left(A_2\right)=0.60$. It is also known that $P\left(A_1 \cap A_2\right)=0$. Suppose $P\left(B \mid A_1\right)=0.20$ and $P\left(B \mid A_2\right)=0.05$.6
a. Are $A_1$ and $A_2$ mutually exclusive? Explain.
b. Compute $P\left(A_1 \cap B\right)$ and $P\left(A_2 \cap B\right)$.
c. Compute $P(B)$.
d. Apply Bayes' theorem to compute $P\left(A_1 \mid B\right)$ and $P\left(A_2 \mid B\right)$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:52

Problem 13

COVID-19 Testing. More than 1.5 million tests for the COVID virus were performed each day in the United States during the COVID-19 pandemic in 2021. Many employers, schools, and government agencies required negative COVID-19 tests before allowing people on their premises. Suppose that for a particular COVID-19 test, the probability of a false negative (meaning that a person who is actually infected with the COVID-19 virus tests negative for the virus) is 0.1 , and that the test always gives a negative result for someone who is not infected with COVID-19. Suppose also the probability that a randomly selected person is infected with COVID-19 is 0.05. Answer the following questions.
a. What is the probability that someone who tests negative for the COVID-19 virus actually has COVID-19?
b. Suppose that a large school system states that 2,500 of its students have reported a negative COVID-19 test. Of these 2,500 students who tested negative, how many would you expect to have COVID-19?
c. Suppose the incidence of COVID-19 is actually considerably higher and the probability of any randomly selected person having COVID-19 is 0.20 . What is the probability in this case that someone who tests negative for the COVID-19 virus actually has COVID-19? Explain this change from part (a).

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:31

Problem 14

Credit Card Defaults. A local bank reviewed its credit-card policy with the intention of recalling some of its credit cards. In the past, approximately $5 \%$ of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of 0.05 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is 0.20 for customers who do not default. Of course, the probability of missing a monthly payment for those who default is 1 .
a. Given that a customer missed a monthly payment, compute the posterior probability that the customer will default.
b. The bank would like to recall its credit card if the probability that a customer will default is greater than 0.20 . Should the bank recall its credit card if the customer misses a monthly payment? Why or why not?

Nick Johnson
Nick Johnson
Numerade Educator
04:51

Problem 15

Prostate Cancer Screening. According to an article in Esquire magazine, approximately $70 \%$ of males over age 70 will develop cancerous cells in their prostate. Prostate cancer is second only to skin cancer as the most common form of cancer for males in the United States. One of the most common tests for the detection of prostate cancer is the prostate-specific antigen (PSA) test. However, this test is known to have a high false-positive rate (tests that come back positive for cancer when no cancer is present). Suppose there is a 0.02 probability that a male patient has prostate cancer before testing. The probability of a false-positive test is 0.75 , and the probability of a falsenegative (no indication of cancer when cancer is actually present) is 0.20 .
a. What is the probability that the male patient has prostate cancer if the PSA test comes back positive?
b. What is the probability that the male patient has prostate cancer if the PSA test comes back negative?
c. For older men, the prior probability of having cancer increases. Suppose that the prior probability of the male patient is 0.3 rather than 0.02 . What is the probability that the male patient has prostate cancer if the PSA test comes back positive? What is the probability that the male patient has prostate cancer if the PSA test comes back negative?
d. What can you infer about the PSA test from the results of parts (a), (b), and (c)?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
07:08

Problem 16

Finding Oil in Alaska. An oil company purchased an option on land in Alaska. Preliminary geologic studies assigned the following prior probabilities.
$$
\begin{aligned}
P(\text { high-quality oil }) & =0.50 \\
P(\text { medium-quality oil }) & =0.20 \\
P(\text { no oil }) & =0.30
\end{aligned}
$$
a. What is the probability of finding oil?
b. After 200 feet of drilling on the first well, a soil test is taken. The probabilities of finding the particular type of soil identified by the test are as follows.
$$
\begin{aligned}
P(\text { soil } \mid \text { high-quality oil }) & =0.20 \\
P(\text { soil I medium-quality oil }) & =0.80 \\
P(\text { soil } \mid \text { no oil }) & =0.20
\end{aligned}
$$

How should the firm interpret the soil test? What are the revised probabilities, and what is the new probability of finding oil?

Samriddhi Singh
Samriddhi Singh
Numerade Educator
04:50

Problem 17

Unemployment Data. Suppose the following data represent the number of persons unemployed for a given number of months in Killeen, Texas. The values in the first column show the number of months unemployed and the values in the second column show the corresponding number of unemployed persons.
$$
\begin{array}{cc}
\text { Months Unemployed } & \text { Number Unemployed } \\
1 & 1,029 \\
2 & 1,686 \\
3 & 2,269 \\
4 & 2,675 \\
5 & 3,487 \\
6 & 4,652 \\
7 & 4,145 \\
8 & 3,587 \\
9 & 2,325 \\
10 & 1,120
\end{array}
$$
Let $x$ be a random variable indicating the number of months a randomly selected person is unemployed.
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 person is unemployed for two months or less? Unemployed for more than two months?
d. What is the probability that a person is unemployed for more than six months?

Andrew Kim
Andrew Kim
Numerade Educator
04:36

Problem 18

Information Systems Job Satisfaction. 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).
$$
\begin{array}{ccc}
\begin{array}{c}
\text { Job Satisfaction } \\
\text { Score }
\end{array} & \begin{array}{c}
\text { IS Senior } \\
\text { Executives (\%) }
\end{array} & \begin{array}{c}
\text { IS Middle } \\
\text { Managers (\%) }
\end{array} \\
1 & 5 & 4 \\
2 & 9 & 10 \\
3 & 3 & 12 \\
4 & 42 & 46 \\
5 & 41 & 28
\end{array}
$$
a. Develop a probability distribution for the job satisfaction score of a randomly selected senior executive.
b. Develop a probability distribution for the job satisfaction score of a randomly selected middle manager.
c. What is the probability that a randomly selected senior executive will report a job satisfaction score of 4 or 5 ?
d. What is the probability that a randomly selected middle manager is very satisfied?
e. Compare the overall job satisfaction of senior executives and middle managers.

Michael Nartey
Michael Nartey
Numerade Educator
02:06

Problem 19

Expectation and Variance of a Random Variable. The following table provides a probability distribution for the random variable $\mathrm{y}$.
a. Compute $E(y)$.
b. Compute $\operatorname{Var}(y)$ and $\sigma$.
$$
\begin{array}{ll}
\boldsymbol{y} & \boldsymbol{f}(\boldsymbol{y}) \\
2 & 0.20 \\
4 & 0.30 \\
7 & 0.40 \\
8 & 0.10
\end{array}
$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator

Problem 20

Financial Statement Audits. Internal auditors are often used to review an organization's financial statements such as balance sheets, income statements, and cash flow statements prior to public filings. Auditors seek to verify that the financial statements accurately represent the financial position of the organization and that the statements follow accepted accounting principles. Many errors that are discovered by auditors are minor errors that are easily corrected. However, some errors are serious and require substantial time to rectify. Suppose that the financial statements of 567 public companies are audited. The file internalaudit contains the number of errors discovered during the internal audit of each of these 567 public companies that were classified as "serious" errors. Use the data in the file internalaudit to answer the following.
a. Construct an empirical discrete probability distribution for the number of serious errors discovered during the internal audits of these 567 public companies.
b. What is the probability that a company has no serious errors in its financial statements?
c. What is the probability that a company has four or more serious errors in its financial statements?
d. What is the expected number of serious errors in a company's financial statements?
e. What is the variance of the number of serious errors in a company's financial statements?
f. What is the standard deviation of the number of serious errors in a company's financial statements?

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01:25

Problem 21

Damage Claims at an Insurance Company. The probability distribution for damage claims paid by the Newton Automobile Insurance Company on collision insurance is as follows.
$$
\begin{array}{rc}
\text { Payment (\$) } & \text { Probability } \\
0 & 0.85 \\
500 & 0.04 \\
1,000 & 0.04 \\
3,000 & 0.03 \\
5,000 & 0.02 \\
8,000 & 0.01 \\
10,000 & 0.01
\end{array}
$$
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? (Hint: It is the expected payments from the company minus the cost of coverage.) Why does the policyholder purchase a collision policy with this expected value?

Nick Johnson
Nick Johnson
Numerade Educator
04:17

Problem 22

Plant Expansion Decision. 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 mediumor 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 $0.20,0.50$, and 0.30 , 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.
(TABLE CAN'T COPY)
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
Numerade Educator
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Problem 23

Binomial Distribution Calculations. Consider a binomial experiment with $n=10$ and $p=0.10$.
a. Compute $f(0)$.
b. Compute $f(2)$.
c. Compute $P(x \leq 2)$.
d. Compute $P(x \geq 1)$.
e. Compute $E(x)$.
f. Compute $\operatorname{Var}(x)$ and $\sigma$.

Shu Naito
Shu Naito
Numerade Educator
04:05

Problem 24

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 tested.
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
01:35

Problem 25

Dog Food Marketing Focus Group. According to the American Veterinary Medical Association (AVMA), $38.4 \%$ of households in the United States own a dog as a pet (AVMA website). Suppose that a company that sells dog food would like to establish a focus group to gather input on a new dog food marketing campaign. The company plans to contact 25 randomly selected households to invite people to join the focus group.
a. Compute the probability that 10 of these 25 households own a dog as a pet.
b. Compute the probability that 2 or fewer of these 25 households own a dog as a pet.
c. For the sample of 25 households, compute the expected number of households who own a dog as a pet.
d. For the sample of 25 households, compute the variance and standard deviation of households who own a dog as a pet.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
05:02

Problem 26

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 two or fewer students will withdraw.
b. Compute the probability that exactly four students will withdraw.
c. Compute the probability that more than three students will withdraw.
d. Compute the expected number of student withdrawals.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:14

Problem 27

Poisson Distribution Calculations. Consider a Poisson distribution with $\mu=3$.
a. Write the appropriate Poisson probability mass function. LO 10
b. Compute $f(2)$.
c. Compute $f(1)$.
d. Compute $P(x \geq 2)$.

Foster Wisusik
Foster Wisusik
Numerade Educator
06:11

Problem 28

Computer Code Errors. The book Code Complete by Steve McDonnell estimates that there are 15 to 50 errors per 1,000 lines of delivered code for computer programs. Assume that for a particular software package, the error rate is 25 per 1,000 lines of code and that the number of errors per 1,000 lines of code follows a Poisson distribution. LO 10
a. What is the probability that a portion of code for this software package that contains 250 lines of code contains no errors?
b. What is the probability of there being exactly 30 errors in 1,000 lines of code?
c. What is the probability of more than 3 errors in 100 lines of code?
d. What is the expected number of errors in a portion of code that contains 25,000 lines of code?

Sonam Khatri
Sonam Khatri
Numerade Educator
02:42

Problem 29

Emergency 911 calls in a large metropolitan area come in at the rate of one every 2 minutes. Assume that the number of 911 calls is a random variable that can be described by the Poisson distribution. LO 10
a. What is the expected number of 911 calls in 1 hour?
b. What is the probability of three 911 calls in 5 minutes?
c. What is the probability of no 911 calls during a 5 -minute period?

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 30

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 and the number of failures follows a Poisson distribution, what is the probability that exactly 4 small businesses will fail during a given month?

Jason Gerber
Jason Gerber
Numerade Educator
03:01

Problem 31

Uniform Distribution Calculations. The random variable $x$ is known to be uniformly distributed between 10 and 20 .
a. Show the graph of the probability density function.
b. Compute $P(x<15)$.
c. Compute $P(12 \leq x \leq 18)$.
d. Compute $E(x)$.
e. Compute $\operatorname{Var}(x)$.

Foster Wisusik
Foster Wisusik
Numerade Educator
03:13

Problem 32

Most computer languages include a function that can be used to generate random numbers. In Excel, the RAND function can be used to generate random numbers between 0 and 1 . If we let $x$ denote a random number generated using RAND, then $x$ is a continuous random variable with the probability density function that follows.
$$
f(x)= \begin{cases}1 & \text { for } 0 \leq x \leq 1 \\ 0 & \text { elsewhere }\end{cases}
$$
a. Graph the probability density function.
b. What is the probability of generating a random number between 0.25 and 0.75 ?
c. What is the probability of generating a random number with a value less than or equal to 0.30 ?
d. What is the probability of generating a random number with a value greater than 0.60 ?
e. Generate 50 random numbers by entering $=$ RAND() into 50 cells of an Excel worksheet.
f. Compute the mean and standard deviation for the random numbers in part (e).

Foster Wisusik
Foster Wisusik
Numerade Educator
02:47

Problem 33

Bidding on a Piece of Land. Suppose we are interested in bidding on a piece of land and we know one other bidder is interested. The seller announced that the highest bid in excess of $$\$ 10,000$$ will be accepted. Assume that the competitor's bid $x$ is a random variable that is uniformly distributed between $$\$ 10,000$$ and $$\$ 15,000$$.
a. Suppose you bid $$\$ 12,000$$. What is the probability that your bid will be accepted?
b. Suppose you bid $$\$ 14,000$$. What is the probability that your bid will be accepted?
c. What amount should you bid to maximize the probability that you get the property?
d. Suppose you know someone who is willing to pay you $$\$ 16,000$$ for the property. Would you consider bidding less than the amount in part (c)? Why or why not?

Kari Hasz
Kari Hasz
Numerade Educator
04:18

Problem 34

Triangular Distribution Calculations. A random variable has a triangular probability density function with $a=50, b=375$, and $m=250$.
a. Sketch the probability distribution function for this random variable. Label the points $a=50, b=375$, and $m=250$ on the $x$-axis.
b. What is the probability that the random variable will assume a value between 50 and 250 ?
c. What is the probability that the random variable will assume a value greater than 300 ?

Amany Waheeb
Amany Waheeb
Numerade Educator
04:29

Problem 35

Project Completion Time. The Siler Construction Company is about to bid on a new industrial construction project. To formulate their bid, the company needs to estimate the time required for the project. Based on past experience, management expects that the project will require at least 24 months, and could take as long as 48 months if there are complications. The most likely scenario is that the project will require 30 months. Assume that the actual time for the project can be approximated using a triangular probability distribution.
a. What is the probability that the project will take less than 30 months?
b. What is the probability that the project will take between 28 and 32 months?
c. To submit a competitive bid, the company believes that if the project takes more than 36 months, then the company will lose money on the project. Management does not want to bid on the project if there is greater than a $25 \%$ chance that they will lose money on this project. Should the company bid on this project?

Amany Waheeb
Amany Waheeb
Numerade Educator
02:15

Problem 36

Large-Cap Stock Fund Returns. Suppose that the return for a particular large-cap stock fund is normally distributed with a mean of $14.4 \%$ and standard deviation of $4.4 \%$. LO 13
a. What is the probability that the large-cap stock fund has a return of at least $20 \%$ ?
b. What is the probability that the large-cap stock fund has a return of $10 \%$ or less?

Nick Johnson
Nick Johnson
Numerade Educator
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Problem 37

A person must score in the upper $2 \%$ of the population on an IQ test to qualify for membership in Mensa, the international high IQ society. If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15 , what score must a person have to qualify for Mensa?

WF
William Fairburn
Numerade Educator
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Problem 38

Amount of Sleep. The United States Centers for Disease Control and Prevention (CDC) recommends that adults sleep seven to nine hours per night (CDC.gov website). However, many adults in the United States sleep less than seven hours per night. Suppose that the amount of sleep per night for an adult in the United States follows a normal distribution with a mean of 8.0 hours and standard deviation of 1.7 hours.
a. What is the probability that a randomly selected adult in the United States sleeps less than 7.0 hours per night?
b. What is the probability that a randomly selected adult in the United States sleeps between 7.0 and 9.0 hours per night?
c. How many hours per night would an adult in the United States sleep if they are in the 10 th percentile of this distribution?

Jason Gerber
Jason Gerber
Numerade Educator
04:00

Problem 39

Assume that the traffic to the web site of Smiley's People, Inc., which sells customized T-shirts, follows a normal distribution, with a mean of 4.5 million visitors per day and a standard deviation of 820,000 visitors per day.
a. What is the probability that the web site has fewer than 5 million visitors in a single day?
b. What is the probability that the web site has 3 million or more visitors in a single day?
c. What is the probability that the web site has between 3 million and 4 million visitors in a single day?
d. Assume that $85 \%$ of the time, the Smiley's People web servers can handle the daily web traffic volume without purchasing additional server capacity. What is the amount of web traffic that will require Smiley's People to purchase additional server capacity?

Yingtai Xiao
Yingtai Xiao
Numerade Educator
05:56

Problem 40

Probability of Defect. Suppose that Motorola uses the normal distribution to determine the probability of defects and the number of defects in a particular production process. Assume that the production process manufactures items with a mean weight of 10 ounces.
a. Assume that the process standard deviation is 0.15 , and the process control is set at plus or minus one standard deviation. Units with weights less than 9.85 or greater than 10.15 ounces will be classified as defects. Calculate the probability of a defect and the expected number of defects for a 1,000 -unit production run.
b. Assume that through process design improvements, the process standard deviation can be reduced to 0.05 . Assume that the process control remains the same, with weights less than 9.85 or greater than 10.15 ounces being classified as defects. Calculate the probability of a defect and the expected number of defects for a 1,000 -unit production run.
c. What is the advantage of reducing process variation, thereby causing process control limits to be at a greater number of standard deviations from the mean?

Andrew Kim
Andrew Kim
Numerade Educator
08:09

Problem 41

Consider the exponential probability density function that follows.
$$
f(x)=\frac{1}{3} e^{-x / 3} \quad \text { for } x \geq 0
$$
a. Write the formula for $P\left(x \leq x_0\right)$.
b. Find $P(x \leq 2)$.
c. Find $P(x \geq 3)$.
d. Find $P(x \leq 5)$.
e. Find $P(2 \leq x \leq 5)$.

Michael Nartey
Michael Nartey
Numerade Educator
02:20

Problem 42

The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds.
a. Sketch this exponential probability distribution.
b. What is the probability that the arrival time between vehicles is 12 seconds or less?
c. What is the probability that the arrival time between vehicles is 6 seconds or less?
d. What is the probability of 30 or more seconds between vehicle arrivals?

Andrew Kim
Andrew Kim
Numerade Educator
04:08

Problem 43

Suppose that the time spent by players in a single session on the World of Warcraft multiplayer online role-playing game follows an exponential distribution with a mean of 38.3 minutes.
a. Write the exponential probability distribution function for the time spent by players on a single session of World of Warcraft.
b. What is the probability that a player will spend between 20 and 40 minutes on a single session of World of Warcraft?
c. What is the probability that a player will spend more than 1 hour on a single session of World of Warcraft?

Wendi Zhao
Wendi Zhao
Numerade Educator
02:01

Problem 44

Laffy Taffy is a type of taffy candy made from corn syrup, sugar, palm oil, and other ingredients. Laffy Taffy comes in a variety of flavors including strawberry, banana, and cherry. Suppose that Laffy Taffy is produced as a continuous length of taffy on an extrusion machine. Defects can occur in the taffy including large air bubbles and tears. Assume that the distance between such defects follows an exponential distribution with a mean of 523 meters.
a. What is the probability that the machine will produce more than 1,000 meters of taffy before the next defect is observed?
b. What is the probability that the next defect will be observed before the machine produces 500 meters or less of taffy?

Sneha Ravi
Sneha Ravi
Numerade Educator