Midterm Practice
Data on U.S. work-related fatalities by cause follow (The World Almanac, 2012).
Assume that a fatality will be randomly chosen from this population. Round your answers to four decimal places. a. What is the probability the fatality resulted from a fall? .1422 b. What is the probability the fatality resulted from a transportation incident? .3958 c. What cause of fatality is least likely to occur? What is the probability the fatality resulted from this cause? .0249
A company that manufactures toothpaste is studying five different package designs.
Assuming that one design is just as likely to be selected by a consumer as any other design, what selection probability would you assign to each of the package designs (to 2 decimals)? .50 In an actual study, 100 consumers were asked to pick the design they preferred. The following data were obtained. Excel File: data04-13.xls Do the data confirm the belief that one design is just as likely to be selected as another? Based on the actual study, what probability would you assign to each package design (to 2 decimals)? Design Probability 1 .05 2 .15 3 .30 4 .40 5 .10 Which package design has the highest probability of selection by a customer? 4
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. How many experimental outcomes exist for the blood analysis experiment? 6 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 Value of x 1 step 1 step 2 1 step 2 steps 3 1 step 3 steps 4 2 steps 1 step 3 2 steps 2 steps 4 2 steps 3 steps 5
Suppose that we have a sample space S = {E1, E2, E3, E4, E5, E6, E7}, where E1, E2, ..., E7 denote the sample points. The following probability assignments apply: P(E1) = 0.1, P(E2) = 0.2, P(E3) = 0.1, P(E4) = 0.2, P(E5) = 0.1, P(E6) = 0.1, and P(E7) = 0.2. Assume the following events when answering the questions.
Find P(A), P(B), and P(C). P(A) .4 P(B) .6 P(C) .6 What is P(A B)? .8 What is P(A B)? .2 Are events A and C mutually exclusive? What is P(Bc )? .4
The National Sporting Goods Association conducted a survey of persons 7 years of age or older about participation in sports activities (Statistical Abstract of the United States: 2002). The total population in this age group was reported at 248.5 million, with 120.9 million male and 127.6 million female. The number of participants for the top five sports activities appears here.
For a randomly selected female, estimate the probability of participation in each of the sports activities (to 2 decimals). Note that the probabilities do not sum to 1 because of participation in more than one sports activity. Bicycle riding .09 Camping .10 Exercise walking .23 Exercising with equipment .10 Swimming .14 For a randomly selected male, estimate the probability of participation in each of the sports activities (to 2 decimals). Note that the probabilities do not sum to 1 because of participation in more than one sports activity. Bicycle riding .10 Camping .10 Exercise walking .11 Exercising with equipment .08 Swimming .10 For a randomly selected person, what is the probability the person participates in exercise walking (to 2 decimals)? .34 Suppose you just happen to see an exercise walker going by. What is the probability the walker is a woman (to 2 decimals)? .68 What is the probability the walker is a man (to 2 decimals)? .32
A random experiment with three outcomes has been repeated 50 times, and it was learned that E1 occurred 20 times, E2 occurred 11 times, and E3 occurred 19 times. Assign probabilities to the following outcomes for E1, E2 and E3. Round your answer to two decimal places.
P(E1) .20 P(E2) .22 P(E3) .38 What method did you use? RELATIVE FREQUENCY
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 follows. Undergraduate Major Business Engineering Other Totals Intended Enrollment Full Time 421 394 74 889 Status Part Time 401 595 46 1,042 Totals 822 989 120 1,931
Use the marginal probabilities of undergraduate major (Business, Engineering, or Other) to comment on which undergraduate major produces the most potential MBA students. Engineering 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 (to 3 decimals)? .443 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 (to 3 decimals)? .512 Let A denote the event that 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 A and B independent? No, they are not independent
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).
What is the expected value of the job satisfaction score for senior executives (to 2 decimals)? 4.05 What is the expected value of the job satisfaction score for middle managers (to 2 decimals)? 3.84 Compute the variance of job satisfaction scores for executives and middle managers (to 2 decimals). Executives 1.25 Middle managers 1.13 Compute the standard deviation of job satisfaction scores for both probability distributions (to 2 decimals). Executives 1.12 Middle managers 1.06 What comparison can you make about the job satisfaction of senior executives and middle managers?
A company studied the number of lost-time accidents occurring at its Brownsville, Texas, plant. Historical records show that 10% of the employees suffered lost-time accidents last year. Management believes that a special safety program will reduce such accidents to 7% during the current year. In addition, it estimates that 15% of employees who had lost-time accidents last year will experience a lost-time accident during the current year.
What is the probability an employee will experience a lost-time accident in both years (to 3 decimals)? .015 What is the probability an employee will experience a lost-time accident over the two-year period (to 3 decimals)? .155
Suppose N = 15 and r = 4.
What is the probability of x = 3 for n = 10 (to 4 decimals)? .4396
A survey of magazine subscribers showed that 45.8% rented a car during the past 12 months for business reasons, 52% rented a car during the past 12 months for personal reasons, and 30% rented a car during the past 12 months for both business and personal reasons. Round your answers to three decimal places.
What is the probability that a subscriber rented a car during the past 12 months for business or personal reasons? .678 b. What is the probability that a subscriber did not rent a car during the past 12 months for either business or personal reasons? .322
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). .000045 b. Compute the probability that three or fewer passengers arrive in a one-minute period (to 4 decimals). .0103 c. Compute the probability of no arrivals in a 15-second period (to 4 decimals). .846 d. Compute the probability of at least one arrival in a 15-second period (to 4 decimals). .154
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, August 6, 2012).
a. Develop a probability distribution for x. x f(x) 0 .3600 1 .2604 2 .1903 3 .0897 4 .0996 b. Compute the expected value of x. 255.0808 cups of coffee c. Compute the variance of x. 31343.8676 cups of coffee squared d. Suppose we are only interested in adults that 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. 193.2619 Compare it to the expected value of x. The input in the box below will not be graded, but may be reviewed and considered by your instructor.
A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately 8% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of .08 that any particular cardholder will default. The bank also found that the probability of missing a monthly payment is .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 one or more monthly payments, compute the posterior probability that the customer will default (to 2 decimals). .303 b. The bank would like to recall its card if the probability that a customer will default is greater than .20. Should the bank recall its card if the customer misses a monthly payment? Why or why not? Yes, the probability is greater than .20
Many students accumulate debt by the time they graduate from college. Shown in the following table is the percentage of graduates with debt and the average amount of debt for these graduates at four universities and four liberal arts colleges. University % with Debt Amount($) College % with Debt Amount($) 1 76 32,920 1 84 28,758 2 69 32,180 2 98 26,000 3 57 11,224 3 57 10,204 4 61 11,854 4 49 11,013
a. If you randomly choose a graduate of College 2, what is the probability that this individual graduated with debt (to 2 decimals)? .98 b. If you randomly choose one of these eight institutions for a follow-up study on student loans, what is the probability that you will choose an institution with more than 50% of its graduates having debt (to 3 decimals)? .875 c. If you randomly choose one of these eight institutions for a follow-up study on student loans, what is the probability that you will choose an institution whose graduates with debts have an average debt of more than $ 20,000 (to 3 decimals)? .500 d. What is the probability that a graduate of University 1 does not have debt (to 2 decimals)? .28 e. For graduates of University 1 with debt, the average amount of debt is $ 32,920. Considering all graduates from University 1, what is the average debt per graduate? Round to nearest dollar. $ 23702
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? Explain. The input in the box below will not be graded, but may be reviewed and considered by your instructor. binomial experiments are repeated trials b. What is the probability that none of the 10 employees will say their company is loyal to them? .0563 c. What is the probability that 4 of the 10 employees will say their company is loyal to them? .1460 d. What is the probability that at least 2 of the 10 employees will say their company is loyal to them? .7560
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 six of the days two were used, on eight of the days three were used, and on three days all four of the hospital's operating rooms were used. Round your answers to two decimal places.
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. x f(x) 1 .15 2 .30 3 .40 4 .15 Total 20 b. Select a graph of the probability distribution. 1. 2. 3. 4. Choose the correct graph from the above Graphs: c. Show that your probability distribution satisfies the required conditions for a valid discrete probability distribution. Because f(x) 0 for x = 1, 2, 3, 4 and sum f(1) + f(2) + f(3) + f(4) = 1.
Twenty-three percent of automobiles are not covered by insurance (CNN, February 23, 2006). On a particular weekend, 35 automobiles are involved in traffic accidents.
a. What is the expected number of these automobiles that are not covered by insurance (0 decimals)? 8 b. What are the variance and standard deviation? Variance 6.2 (to 1 decimal) Standard deviation 2.49 (to 2 decimals)
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)? .0489 b. What is the probability of at least two off-the-job accidents during a one-year period (to 4 decimals)? .8008 c. What is the expected number of off-the-job accidents during six months (to 1 decimal)? 1.5 d. What is the probability of no off-the-job accidents during the next six months (to 4 decimals)? .2231
A decision maker subjectively assigned the following probabilities to the four outcomes of a random experiment: P(E1) = .10, P(E2) = .15, P(E3) = .40, and P(E4) = .20. Are these probability assignments valid?
no they are greater than 0 but do not equal 1
A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take 1.1, 2.3, 3.3, or 4.2 hours. The different types of malfunctions occur at the same frequency. If required, round your answers to two decimal places. Develop a probability distribution for the duration of a service call.
Duration of Call x f(x) 1.1 .10 2.3 .21 3.3 .30 4.2 .39 1 Which of the following probability distribution graphs accurately represents the data set? Consider the required conditions for a discrete probability function, shown below. Does this probability distribution satisfy equation (5.1)? Does this probability distribution satisfy equation (5.2)? What is the probability a randomly selected service call will take 3.3 hours? .30 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 P.M. What is the probability the service technician will have to work overtime to fix the machine today? .82
