CSU Probability Math Equations Worksheet

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Running Head: MASTERY EXERCISE 1 Mastery Exercise 4 Q1) 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 onethird 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 (to 4 decimals)? b. What is the probability that one of the meals will exceed the cost covered by your company (to 4 decimals)? c. What is the probability that two of the meals will exceed the cost covered by your company (to 4 decimals)? d. What is the probability that all three of the meals will exceed the cost covered by your company (to 4 decimals)? Q2) A Randstad/Harris interactive survey reported that 25% of employees said their company is loyal to them (USA Today, November 11, 2009). Suppose 10 employees are selected randomly and will be interviewed about company loyalty. If required, round your answers to four decimal places. a. Is the selection of 10 employees a binomial experiment? - Select your answ er - Explain. The input in the box below will not be graded, but may be reviewed and considered by your instructor. b. What is the probability that none of the 10 employees will say their company is loyal to them? Running Head: MASTERY EXERCISE 2 c. What is the probability that 4 of the 10 employees will say their company is loyal to them? d. What is the probability that at least 2 of the 10 employees will say their company is loyal to them? Q3) 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 (to 6 decimals). b. Compute the probability that three or fewer passengers arrive in a one-minute period (to 4 decimals). c. Compute the probability of no arrivals in a 15-second period (to 4 decimals). d. Compute the probability of at least one arrival in a 15-second period (to 4 decimals). Q4) Airline passengers arrive randomly and independently at the passenger-screening facility at a major international airport. The mean arrival rate is 6 passengers per minute. a. Compute the probability of no arrivals in a one-minute period (to 6 decimals). b. Compute the probability that three or fewer passengers arrive in a one-minute period (to 4 decimals). c. Compute the probability of no arrivals in a 15-second period (to 4 decimals). d. Compute the probability of at least one arrival in a 15-second period (to 4 decimals). Running Head: MASTERY EXERCISE 3 Q5) The following probability distributions of job satisfaction scores for a sample of information systems (IS) senior executives and IS middle managers range from a low of 1 (very dissatisfied) to a high of 5 (very satisfied). Excel File: data05-21.xls a. What is the expected value of the job satisfaction score for senior executives (to 2 decimals)? b. What is the expected value of the job satisfaction score for middle managers (to 2 decimals)? c. Compute the variance of job satisfaction scores for executives and middle managers (to 2 decimals). Executives Middle managers d. e. Compute the standard deviation of job satisfaction scores for both probability distributions (to 2 decimals). Executives Middle managers f. g. What comparison can you make about the job satisfaction of senior executives and middle managers? - Select your answ er - Q6) West Virginia has one of the highest divorce rates in the nation, with an annual rate of approximately 5 divorces per 1000 people (Centers for Disease Control and Prevention website, January 12, 2012). The Marital Counseling Center, Inc. (MCC) thinks that the high divorce rate in the state may require them to hire additional staff. Working with a consultant, the management of Running Head: MASTERY EXERCISE 4 MCC has developed the following probability distribution for x = the number of new clients for marriage counseling for the next year. Excel File: data05-19.xls a. Is this probability distribution valid? - Select your answ er - Explain. f(x) - Select your answ er - Σf(x) - Select your answ er - b. What is the probability MCC will obtain more than 30 new clients (to 2 decimals)? c. What is the probability MCC will obtain fewer than 20 new clients(to 2 decimals)? d. Compute the expected value and variance of x. Expected value clients per year Variance squared clients per year Q7) The probability distribution for the random variable x follows. x a. f(x) 20 0.23 24 0.11 31 0.27 35 0.39 Is this a valid probability distribution? Select b. What is the probability that x = 31 (to 2 decimals)? c. What is the probability that x is less than or equal to 24 (to 2 decimals)? Running Head: MASTERY EXERCISE d. 5 What is the probability that x is greater than 31 (to 2 decimals)? Q8) The unemployment rate in the state of Arizona is 4.1% (CNN Money website, May 2, 2007). Assume that 100 employable people in Arizona are selected randomly. a. What is the expected number of people who are unemployed (to 1 decimal)? b. What are the variance and standard deviation of the number of people who are unemployed (to 2 decimals)? Variance Standard deviation Q9) 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 an All World fund composed of global stocks with good dividend yields. One of their clients is interested in a portfolio consisting of investment 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? - Select your answ er - Why? The input in the box below will not be graded, but may be reviewed and considered by your instructor. 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? Running Head: MASTERY EXERCISE 6 Expected return (Round your answer to three decimal places) Standard deviation (Round your answer to two decimal places) What would be the expected return and standard deviation, in dollars, for a client investing $10,000 in such a portfolio? Expected return Standard deviation $ $ 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? Round your answer to three decimal places. Expected return (Round your answer to three decimal places) Standard deviation (Round your answer to four decimal places) What would be the expected return and standard deviation, in dollars, for a client investing $10,000 in such a portfolio? Expected return Standard deviation $ $ d. Which of the portfolios in parts (b) and (c) would you recommend for an aggressive investor? - Select your answ er - Which would you recommend for a conservative investor? - Select your answ er - Why? The input in the box below will not be graded, but may be reviewed and considered by your instructor. Running Head: MASTERY EXERCISE 7 Q10) 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. a. Consider the required conditions for a discrete probability function, shown below. Does this probability distribution satisfy equation (5.1)? - Select your answ er - Does this probability distribution satisfy equation (5.2)? - Select your answ er - b. What is the probability that it takes exactly 2 sessions to gain the patient's trust (to 3 decimals)? c. What is the probability that it takes at least 2 sessions to gain the patient's trust (to 3 decimals)? Q11) Video Link assistance: https://www.cengage.com/decisionsciences/book_content/anderson_9781133274537/video/video.ht ml?asset=chpt5-31 Consider a binomial experiment with two trials and p =0.9. a. Which of the following tree diagrams accurately represents this binomial experiment? Running Head: MASTERY EXERCISE b. Select c. Compute the probability of one success, f(1) (to 2 decimals). d. Compute f(0) (to 2 decimals). e. Compute f(2) (to 2 decimals). f. Compute the probability of at least one success (to 2 decimals). g. Compute the following (to 2 decimals). 8 Running Head: MASTERY EXERCISE Expected value Variance Standard deviation Q12) The National Safety Council (NSC) estimates that off-the-job accidents cost U.S. businesses almost $200 billion annually in lost productivity (National Safety Council, March 2006). Based on NSC estimates, companies with 50 employees are expected to average three employee off-the-job accidents per year. Answer the following questions for companies with 50 employees. a. What is the probability of no off-the-job accidents during a one-year period (to 4 decimals)? b. What is the probability of at least two off-the-job accidents during a one-year period (to 4 decimals)? c. What is the expected number of off-the-job accidents during six months (to 1 decimal)? d. What is the probability of no off-the-job accidents during the next six months (to 4 decimals)? Q13) Three students scheduled interviews for summer employment at an 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. How many experimental outcomes exist? b. Let x equal the number of students who receive an offer. Is x continuous or discrete? Select c. Show the value of the random variable x, where x is the number of yeses. Let Y = "Yes, the student receives an offer", and N = "No, the student does not receive an offer." Experimental Outcome Value of x 9 Running Head: MASTERY EXERCISE 10 (N, N, Y) (Y, N, N) (N, Y, Y) (N, Y, N) (Y, Y, Y) (Y, N, Y) (Y, Y, N) (N, N N) Q14) 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. How many experimental outcomes exist for the blood analysis experiment? b. Let x denote the total number of steps required to do the analysis (both procedures). Show the value of the random variable for each of the experimental outcomes. Experimental Outcomes Procedure 1 Procedure 2 1 step 1 step 1 step 2 steps 1 step 3 steps 2 steps 1 step 2 steps 2 steps 2 steps 3 steps Value of x Running Head: MASTERY EXERCISE Q15) A university found that 30% 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 (to 4 decimals). b. Compute the probability that exactly 4 will withdraw (to 4 decimals). c. Compute the probability that more than 3 will withdraw (to 4 decimals). d. Compute the expected number of withdrawals. 11
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please find attached

Running Head: MASTERY EXERCISE

1
Mastery Exercise 4

Q1)
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 onethird 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 (to 4
decimals)?
0.2637

b. What is the probability that one of the meals will exceed the cost covered by your company (to 4
decimals)?
0.4945

c. What is the probability that two of the meals will exceed the cost covered by your company (to 4
decimals)?
0.2198

d. What is the probability that all three of the meals will exceed the cost covered by your company
(to 4 decimals)?
0.0220

Q2)
A Randstad/Harris interactive survey reported that 25% of employees said their company is loyal to
them (USA Today, November 11, 2009). Suppose 10 employees are selected randomly and will be
interviewed about company loyalty.

If required, round your answers to four decimal places.
a. Is the selection of 10 employees a binomial experiment?
Yes

Explain.
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
Yes, it is a binomial experiment because there are a
Lastly, the probability of success is the same for eac

b. What is the probability that none of the 10 employees will say their company is loyal to them?
0.0563

Running Head: MASTERY EXERCISE

2

c. What is the probability that 4 of the 10 employees will say their company is loyal to them?
0.1460

d. What is the probability that at least 2 of the 10 employees will say their company is loyal to them?
0.7560

Q3)
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 (to 6 decimals).
0.000045

b. Compute the probability that three or fewer passengers arrive in a one-minute period (to 4 decimals).
0.0103

c. Compute the probability of no arrivals in a 15-second period (to 4 decimals).
0.0821

d. Compute the probability of at least one arrival in a 15-second period (to 4 decimals).
0.9179

Q4)
Airline passengers arrive randomly and independently at the passenger-screening facility at a major
international airport. The mean arrival rate is 6 passengers per minute.

a.

Compute the probability of no arrivals in a one-minute period (to 6 decimals).
0.002479

b.

Compute the probability that three or fewer passengers arrive in a one-minute period (to 4 decimals).
0.1512

c.

Compute the probability of no arrivals in a 15-second period (to 4 decimals).
0.2231

d.

Compute the probability of at least one arrival in a 15-second period (to 4 decimals).
0.7769

Running Head: MASTERY EXERCISE

3

Q5)
The following probability distributions of job satisfaction scores for a sample of information systems
(IS) senior executives and IS middle managers range from a low of 1 (very dissatisfied) to a high of 5
(very satisfied).
Excel File: data05-21.xls

a.

What is the expected value of the job satisfaction score for senior executives (to 2 decimals)?
4.05

b.

What is the expected value of the job satisfaction score for middle managers (to 2 decimals)?
3.84

c.

Compute the variance of job satisfaction scores for executives and middle managers (to 2 decimals).
Executives

1.25

Middle managers

1.13

d.
e.

Compute the standard deviation of job satisfaction scores for both probability distributions (to 2
decimals).
Executives

1.12

Middle managers

1.07

f.
g.

What comparison can you make about the job satisfaction of senior executives and middle managers?
Senior executives have higher satisfaction w ith more variation

Q6)
West Virginia has one of the highest divorce rates in the nation, with an annual rate of approximately
5 divorces per 1000 people (Centers for Disease Control and Prevention website, January 12, 2012).
The Marital Counseling Center, Inc. (MCC) thinks that the high divorce rate in the state may require
them to hire additional staff. Working with a consultant, the management of MCC has developed the

Running Head: MASTERY EXERCISE

4

following probability distribution for x = the number of new clients for marriage counseling for the
next year.
Excel File: data05-19.xls

a. Is this probability distribution valid?
Yes

Explain.
f(x)

greater than or equal to 0

Σf(x)

equal to 1

b. What is the probability MCC will obtain more than 30 new clients (to 2 decimals)?
0.75

c. What is the probability MCC will obtain fewer than 20 new clients(to 2 decimals)?
0.05

d. Compute the expected val...


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