Statistics

3. You live in State College, where CATA buses always arrive exactly on time, with the time between successive buses xed at 10 minutes. Having lost your watch, you arrive at the bus stop at a random time (assume that buses run 24 hours a day, and that the time that you arrive is uniformly random on a particular day). (a) (2 pts) What is the distribution of how long you have to wait for the next bus? What is the average time that you have to wait? (b) (2 pts) Given that the bus has not yet arrived after 6 minutes, what is the probability that you will have to wait at least 3 more minutes? 4. The SAT and ACT college entrance exams are taken by thousands of students each year. The mathematics portion of each of these exams produces scores that are approximately Normally distributed. In recent years, SAT mathematics exam scores have averaged 480 with a standard deviation of 100. The average and standard deviation for the ACT mathematics scores are 18 and 6, respectively. (a) (2 pts) Suppose you made a 580 on the mathematics section of the SAT. What is your standard (i.e., standardized) score for the mathematics section? (b) (2 pts) What percent of all students scored below you on the mathematics section of the SAT? (0) (2 pts) What would have been the comparable score on the mathematics section of the ACT? 5. (5 pts) A certain machine often breaks down and needs to be xed. At time 0, the machine is working. It works for an Expo()\\) period of time (measured in days), and then breaks down. It then takes an EXDO()\\) amount of time to get it xed, after which it will work for an Expo(/\\) time until it breaks down again, after which it will take an Expo()\\) time to get it fixed, etc. Assume that these Expo()\\) r.v.s are iii.d. A transition occurs when the machine switches from working to being broken, or switches from being broken to working. Find the distribution of the number of transitions that occur in t days, i.e., within the time interval (0,19%)