BUS277 Exam Review Ch. 3, 4 and 5 In October 2012, Apple introduced a much smaller variant of the Apple iPad, known as the iPad Mini. Weighing less than 11 ounces, it was about 50% lighter than the standard iPad. Battery tests for the iPad Mini showed a mean life of 10.25 hours (The Wall Street Journal, October 31, 2012). Assume that battery life of the iPad Mini is uniformly distributed between 8.5 and 12 hours. a) Give a mathematical expression for the probability density function of battery life. b) c) d) e) The correct answer is: a What is the probability that the battery life for an iPad Mini will be 10 hours or less (to 4 decimals)? 0.4286 What is the probability that the battery life for an iPad Mini will be at least 11 hours (to 4 decimals)? 0.2857 What is the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours (to 4 decimals)? 0.5714 In a shipment of 100 iPad Minis, how many should have a battery life of at least 9 hours (to nearest whole value)? 86 The prior probabilities for events A1 and A2 are P(A1) = .40 and P(A2) = .60. It is also known that P(A1 A2) = 0. Suppose P(B | A1) = .10 and P(B | A2) = .08. a) Are events A1 and A2 mutually exclusive? Yes b) Compute P(A1 B) (to 4 decimals). 0.04 Compute P(A2 B) (to 4 decimals). 0.048 c) Compute P(B) (to 4 decimals). d) Apply Bayes' theorem to compute P(A1 | B) (to 4 decimals). 0.4545 Also apply Bayes' theorem to compute P(A2 | B) (to 4 decimals). 0.5455 A local bank reviewed its credit card policy with the intention of recalling some of its credit cards. In the past approximately 6% of cardholders defaulted, leaving the bank unable to collect the outstanding balance. Hence, management established a prior probability of .06 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). 0.24 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 of default is greater than 0.2 The probability distribution for the random variable x follows. a) b) c) d) Is this a valid probability distribution? Yes, because all f(x) 0 and sum of f(x) = 1 What is the probability that x = 30 (to 2 decimals)? 0.28 What is the probability that x is less than or equal to 25 (to 2 decimals)? 0.32 What is the probability that x is greater than 30 (to 2 decimals)? 0.4 For unemployed persons in the United States, the average number of months of unemployment at the end of December 2009 was approximately seven months (Bureau of Labor Statistics, January 2010). Suppose the following data are for a particular region in upstate New York. 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. c) What is the probability that a person is unemployed for two months or less (to 4 decimals)? 0.1006 Unemployed for more than two months (to 4 decimals)? 0.8994 d) What is the probability that a person is unemployed for more than six months (to 4 decimals)? 0.4143 The following table provides a probability distribution for the random variable y. a) Compute E(y) (to 1 decimal). 5.6 b) Compute Var(y) and (to 2 decimals). Var=4.44 and =2.11 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 following probability distribution for x = the number of new clients for marriage counseling for the next year. a) Is this probability distribution valid? Yes Explain. f(x) 0 f(x) = 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 value and variance of x. Expected value 43 Variance 201