Statistics Free-Response Questions

STATISTICS FREE-RESPONSE QUESTIONS 4. A company manufactures model rockets that require igniters to launch. Once an igniter is used to launch a rocket, the igniter cannot be reused. Sometimes an igniter fails to operate correctly, and the rocket does not launch. The company estimates that the overall failure rate, defined as the percent of all igniters that fail to operate correctly, is 15 percent. A company engineer develops a new igniter, called the super igniter, with the intent of lowering the failure rate. To test the performance of the super ignitors, the engineer uses the following process. Step 1: One super igniter is selected at random and used in a rocket. Step 2: If the rocket launches, another super igniter is selected at random and used in a rocket Step 2 is repeated until the process stops. The process stops when a super igniter fails to operate correctly or 32 super igniters have successfally launched rockets, whichever comes first. Assume that super igniter failures are independent. (a) If the failure rate of the super igniters is 15 percent, what is the probability that the first 30 super igniters selected using the testing process successfully launch rockets? )(1-15) (85)2.0076 (b) Given that the first 30 super ignitets successfully launch rockets, what is the probability that the first failure occurs on the thirty-first or the thirty-second super igniter tested if the failure rate of the super igniters is 15 percent? 15+ (85) (15) .2775 (c) Given that the first 30 super igniters succesfully launch rockets, is it reasonable to believe that the failure rate of the xaper igniters is less than 15 percent? Explain. (c) Given that the first 30 super igniters successfully launch rockets, is it reasonable to believe that the failure rate of the super igniters is less than 15 percent? Explain. c) yes, ble the prob of 30 successes in a row is 0076, which is very unlikely to happen if the failure rate is 15. So, there is enough reason to suggest the failure rate is really less than -15. 3. A shopping mall has three automated teller machines (ATMs), Because the machines receive heavy use, they sometimes stop working and need to be repaired. Let the random variable X represent the number of ATMs that are working when the mall opens on a randomly selected day. The table shows the probability distribution of X. Number of ATMs working when the mall opens D 2 3 Probability 0.15 1 0.21 0.40 0.24 (a) What is the probability that at least one ATM is working when the mall opens? P(X≥1)=0.21+0.40+0.24=0.85. (b) What is the expected value of the number of ATMs that are working when the mall opens? E(X)=0(0.15) +1(0.21)+2(0.40)+3(0.24)=1.73 machines. (c) What is the probability that all three ATMs are working when the mall opens, given that at least one ATM is working? P(X=3X21) P(X=3 and X21) P(X=3) 0.24 P(X 21) P(X≥1) 0.85 0.282 (d) Given that at least one ATM is working when the mall opens, would the expected value of the number of ATMs that are working be less than, equal to, or greater than the expected value from part (b)? Explain. Cliven that at least one ATM is working when the mall opens, the expected value of the number of worlang ATMs would be greater than the expected value calculated in part (b). By eliminating the possibility of 0 working ATMs, the probabilities for 1. 2. and 3 working ATMs all increase proportionally, so the expected value must increase