Download Probability and Hypothesis Testing Problems and more Assignments Mathematics in PDF only on Docsity! Homework Assignment due Tuesday, April 26 Problem 1. Suppose that children are born in a certain hospital at a Poisson rate of six per day. What is the probability that (a) At least two babies are born during the next six hours; (b) No babies are born during the next day? (c) What is the average time between the arrival of babies? (d) Three hours have elapsed since the last baby was born. What’s the probability that a baby will be born in the next hour? Problem 2. Suppose that earthquakes of magnitude 5.5 or higher on the Richter scale have a probability 0.02 of damaging a certain type of bridge. Suppose that such intense earthquakes occur at a Poisson rate of 2 every ten years. If a bridge of this type is constructed to last at least 60 years, what is the probability that it will be undamaged by earthquakes for that period of time? Problem 3. Cars arrive at a parking lot at a Poisson rate of 12 per hour. Find: (a) The probability that no cars arrive in the next 15 minutes. (b) The probability that the first car arrives in the next 15 minutes. (c) After we have waited 15 minutes, no cars have arrived. What’s the probability that the first car arrives in the next fifteen minutes? Problem 4. The number of users at a local ATM follows a Poisson distribution, with a mean of 15 per hour. (a) Find the probability that there will be at least 2 users in the next 10 minutes. (b) What is the average interval of time between two consecutive users? (c) Two minutes have elapsed since the last user left the ATM. What is the probability that at least an additional 5 minutes will elapse before the next user arrives? Problem 5 (Final Exam, Spring 2004). Before 1995, the Dow Jones Industrials (DJI), a key economic indicator, return on average 12% annual growth earnings. Sally, an economics student, suspects that the economy fundamentally changed during the five-year period 1995—1999. Using the annual growth earnings of the DJI during this five year period, she hopes to show this with a test of hypothesis. She assumes that the growth earnings are normally distributed. (a) Formulate the null and alternative hypotheses. (b) Choose an appropriate test statistic and formulate the rejection rule at 10% level of significance. (c) For the period 1995—1999, the yearly returns for the DJI were: 1995 1996 997 1998 1999 33% 26% 23% 16% 25% What conclusion should she reach? Problem 6. You’ve always believed that a bag of N & N’s—a pill-like chocolate candy—always contains 25 pieces of candy, no more, no less. But lately you haven’t been satisfied; you suspect bags that are being sold in your movie theatre have fewer pieces than this. You decide to do a hypothesis test. (a) Formulate the null and alternative hypotheses. 1
