{Exercise 5.11 (Algorithmic)} A technician services mailing machines at companies in the Phoenix area. Depending on the type of malfunction, the service call can take 1.1, 2.2, 3.1, or 4.3 hours. The different types of malfunctions occur at the same frequency. If required, round your answers to two decimal places. 1. Develop a probability distribution for the duration of a service call. Duration of Call x f(x) 1.1 ____ 2.2 ____ 3.1 ____ 4.3 ____ ____ 2. 3. Which of the following probability distribution graphs accurately represents the data set? 4. 5. 6. _________________ 7. 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)? _________________ 8. What is the probability a service call will take 3.1 hours? ____ 9. 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? ____ 2. eBook {Exercise 5.13} 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. 1. 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)? _________________ 2. What is the probability that it takes exactly 2 sessions to gain the patient's trust (to 3 decimals)? ______ 3. What is the probability that it takes at least 2 sessions to gain the patient's trust (to 3 decimals)? ______ 3. eBook {Exercise 5.17} The number of students taking the SAT has risen to an all-time high of more than 1.5 million (College Board, August 26, 2008). Students are allowed to repeat the test in hopes of improving the score that is sent to college and university admission offices. The number of times the SAT was taken and the number of students are as follows. Excel File: data05-17.xls a. Let x be a random variable indicating the number of times a student takes the SAT. Show the probability distribution for this random variable (to 4 decimals). x=1 _________________ x=2 _________________ x=3 _________________ x=4 _________________ x=5 _________________ b. What is the probability that a student takes the SAT more than one time (to 4 decimals)? _________________ c. What is the probability that a student takes the SAT three or more times (to 4 decimals)? _________________ d. What is the expected value of the number of times the SAT is taken (to 4 decimals)? _________________ What is your interpretation of the expected value? The input in the box below will not be graded, but may be reviewed and considered by your instructor. _________________ e. What is the variance and standard deviation for the number of times the SAT is taken (to 4 decimals)? Variance _________________ Standard deviation _________________ 4. eBook {Exercise 5.23} The New York City Housing and Vacancy Survey showed a total of 59,324 rent-controlled housing units and 236,263 rent-stabilized units built in 1947 or later. For these rental units, the probability distributions for the number of persons living in the unit are given (U.S. Census Bureau website, January 12, 2004). Excel File: data05-23.xls a. What is the expected value of the number of persons living in each type of unit (to 2 decimals)? Rent-controlled _________________ Rent-stabilized _________________ b. What is the variance of the number of persons living in each type of unit (to 2 decimals)? Rent-controlled _________________ Rent-stabilized _________________ c. Make some comparisons between the number of persons living in rent-controlled units and the number of persons living in rent-stabilized units. The input in the box below will not be graded, but may be reviewed and considered by your instructor. _________________ 5. eBook {Exercise 5.35 (Algorithmic)} In San Francisco, 30% of workers take public transportation daily (USA Today, December 21,2005). 1. In a sample of 8 workers, what is the probability that exactly three workers take public transportation daily (to 4 decimals)? ______ 2. In a sample of 8 workers, what is the probability that at least three workers take public transportation daily (to 4 decimals)? ______ 6. eBook {Exercise 5.41 (Algorithmic)} A university found that 40% of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course. 1. Compute the probability that 2 or fewer will withdraw (to 4 decimals). ______ 2. Compute the probability that exactly 4 will withdraw (to 4 decimals). ______ 3. Compute the probability that more than 3 will withdraw (to 4 decimals). ______ 4. Compute the expected number of withdrawals. ______ 7. eBook {Exercise 5.47} During the period of time that a local university takes phone-in registrations, calls come in at the rate of one every two minutes. 1. What is the expected number of calls in one hour? ______ 2. What is the probability of three calls in five minutes (to 4 decimals)? ______ 3. What is the probability of no calls in a five-minute period (to 4 decimals)? ______ 8. eBook {Exercise 5.51} 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. 1. What is the probability of no off-the-job accidents during a one-year period (to 4 decimals)? ______ 2. What is the probability of at least two off-the-job accidents during a one-year period (to 4 decimals)? ______ 3. What is the expected number of off-the-job accidents during six months (to 1 decimal)? ______ 4. What is the probability of no off-the-job accidents during the next six months (to 4 decimals)? ______ 9. eBook {Exercise 5.61 (Algorithmic)} 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: 0.25, 0.16, 0.25, 0.13, and 0.21. 1. Show the probability distribution for the expense forecast (to 2 decimals). x f(x) 9 ____ 10 ____ 11 ____ 12 ____ 13 ____ 2. 3. What is the expected value of the expense forecast for the coming year (to 2 decimals)? ______ 4. What is the variance of the expense forecast for the coming year (to 2 decimals)? _______ 5. If income projections for the year are estimated at $12 million, how much profit does the college expect to make (report your answer in millions of dollars, to 2 decimals)? ____ 10. eBook {Exercise 5.65} Apolitical action group is planning to interview home owners to assess the impact caused by a recent slump in housing prices. According to a Wall Street Journal/Harris Interactive Personal Finance poll, 26% of individuals aged 18-34, 50% of individuals aged 35-44, and 88% of individuals aged 55 and over are home owners (All Business website, January 23, 2008). a. How many people from the 18-34 age group must be sampled to find an expected number of at least 20 home owners? _________________ b. How many people from the 35-44 age group must be sampled to find an expected number of at least 20 home owners? _________________ c. How many people from the 55 and over age group must be sampled to find an expected number of at least 20 home owners? _________________ d. If the number of 18-34 year olds sampled is equal to the value identified in part (a), what is the standard deviation of the number who will be home owners (to 2 decimals)? _________________ e. If the number of 35-44 year olds sampled is equal to the value identified in part (b), what is the standard deviation of the number who will be home owners (to 2 decimals)? _________________ 11. eBook {Exercise 5.67} 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 the nearest whole number)? _________________ b. What are the Variance and Standard deviation of the number of people who are unemployed (to 2 decimals)? Variance _________________ Standard deviation _________________ 12. eBook {Exercise 5.69} 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 (to 4 decimals)? ______