ISYE 6644 - Final prep 2023

ISYE 6644 - Final prep 2023 What is a possible goal of an indifference-zone normal means selecti on technique? - ANSWER Find the normal population having the largest mean, e specially if the largest mean is ≫ the second-largest. TRUE or FALSE? The Bechhofer procedure for selecting the normal population with the largest mean specifies the appropriate number of observations to take from each competing population, and simply selects the competitor having the largest sample mean. - ANSWER True TRUE or FALSE? Sometimes a single-stage procedure like Bechhofer's is ine fficient. In fact, it's possible to use certain sequential procedures that take observations one- at-a-time (instead of all at once in a single stage) to make good selection decisions using fewer observations. - ANSWER True For which scenarios(s) below might it be appropriate to use a Bernoulli se lection procedure? a) Find the inventory policy having the largest profit. b) Find the drug giving the best chance of a cure. c) Find the maintenance policy having the lowest failure probabili ty. d) Find the scheduling rule that that has the best chance of maki ng an on-time delivery. - ANSWER All three of (b), (c), and (d). Suppose that a Bernoulli selection procedure tells you to take 100 observa tions from each of two populations, A and B. It turns out that A gets 85 successes and B g ets 46 successes. What do you think? - ANSWER 1) A almost certainly has a higher succe ss probability than B. 2) We could've probably stopped sampling a bit earlier (i.e. , with fewer than 100 observations) because A was so far ahead of B. For which scenarios(s) below might it be appropriate to use a multinomial selection procedure? - ANSWER Find the most-popular political candidat e. Suppose that we want to know which of Coke, Pepsi and Dr.pepp er is the most popular. We would like to make the correct selection with probability of at least P*=0.90 in the event that the ration of the highest- to-second-highest preference probabilities happens to be at least 0*=1.4. How many people does the single-stage procedure Mbem re quire us to interview? - ANSWER 126 Go to the table in the notes and pick off the entry for k=3, P^ ⋆=0.90, and θ^⋆=1.4. Which of the following parameters can you get confidence interval s for? Means Variances Quantiles Differences between the means of two systems - ANSWER All o f the above If we have an iid normal sample of observations, X1, X2,...Xn, what probability distribution is most-commonly used to obtain cofidence intervals for the mean? - ANSWER t TRUE or FALSE? The paired CI for the differences in two means is designed to work especially well if all of the observations from the first population are co mpletely independent of all of the observations from the second population. - ANSWER FALSE. {In fact, it's easier to distinguish between the two means if Xi is positively correl ated with Yi. Think about my parallel parking example in the class notes.} TRUE or FALSE? You can use a version of independent replications to obtain confi dence intervals for the difference in the means from two simulation mode ls. - ANSWER TRUE TRUE or FALSE? The common random numbers technique intentionally i nduces positive correlation between two systems - much like a paired- 𝑡confidence interval. - ANSWER TRUE CRN depends on someone's ability to manipulate the underlying pseudo-ra ndom numbers - e.g., use the same arrival times when simulating two competing si mulated systems. So who ultimately controls those PRNs?? - ANSWER You do - you are p owerful! TRUE or FALSE? The antithetic random numbers technique intentionally induces negative correlation between two runs of the same system - this allows us to better estimate the mean of the system. - ANSWER TRUE TRUE or FALSE? The control variates technique provides unbiased, low-varia nce estimates using a method reminiscent of regression. - ANSWER TRUE Statistical ranking and selection techniques have been designed to address a variety of comparison problems. Which ones from the following list? Find the population having the largest mean. Find the system with the smallest variance. Find the alternative with the highest success probability. Find the most-popular candidate. All of the above. - ANSWER All of the above. Suppose we are dealing with i.i.d. normal observations with un known variance. Which of the following is true about a 95% confidence interval for the mean μ? - ANSWER We are 95% sure that our CI will actually contain the unknown value of μ. We are studying the waiting times arising from two queuei ng systems. Suppose we make 4 independent replications of both systems, where the systems are si mulated independently of each other. replication system 1 system2 1 10 25 2 20 10 3 5 40 4 30 30 Assuming that the average waiting time results from each replicati on are approximately normal, find a two-sided 95% CI for the difference in the me ans of the two systems. - ANSWER This is a two-sample CI problem assuming unknown and unequal variances. We have [-29.76, 9.76] This is sort of the same as Question 2, except we have now used co mmon random numbers to induce positive correlation between the results of the two system s. Again find a two-sided 95% CI for the difference in the means of the two systems. - ANSWER This is a paired-t CI problem assuming unknown variance of the differences. [-16.5, -3.5] Suppose A and B are two identically distributed, unbiased, antith etic estimators for the mean μ of some random variable, and let C = ( A + B ) / 2. Which of the f ollowing is true? - ANSWER E [ C ] = μ and V a r ( C ) < V a r ( A ) / 2. Suppose that you want to pick that one of three normal populations having the largest mean. We'll assume that the variances of the three competitors are all kn own to be equal to σ 2 = 4.