ISYE 6644 - Final prep 2025 graded A with correct Answers

ISYE 6644 - Final prep 2025What is a possible goal of an indifference-zone normal means selection technique? - ANSWER Find the normal population having the largest mean, especially if the largest mean is ≫ the second-largest. TRUE or FALSE? The Bechhofer procedure for selecting the normal population with thelargestmean specifies the appropriate number of observations to take from each competing population, and simply selects the competitor having the largest sample mean. - ANSWERTrue TRUE or FALSE? Sometimes a single-stage procedure like Bechhofer's is inefficient. Infact, it'spossible to use certain sequential procedures that take observations one-at-a-time (insteadof 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 selection 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 probability. d) Find the scheduling rule that that has the best chance of making an on-time delivery. - ANSWER All three of (b), (c), and (d). Suppose that a Bernoulli selection procedure tells you to take 100 observations fromeachof two populations, A and B. It turns out that A gets 85 successes and B gets 46 successes. Whatdo you think? - ANSWER 1) A almost certainly has a higher success probability thanB. 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 candidate. Suppose that we want to know which of Coke, Pepsi and Dr is the most popular. Wewould 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 require us to interview?- ANSWER 126 ONLYSTUDENTS 2025 ALL THE BEST ONLYSTUDENTS STORE 2 DO NOT COPY 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 intervals for? Means Variances Quantiles Differences between the means of two systems - ANSWER All of the above If we have an iid normal sample of observations, X1, X2,...Xn, what probability distributionismost-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 especiallywell if all of the observations from the first population are completely independent of all of the observations from the second population. - ANSWER FALSE. {In fact, it's easier todistinguish between the two means if Xi is positively correlated with Yi. Think about myparallel parking example in the class notes.} TRUE or FALSE? You can use a version of independent replications to obtain confidenceintervals for the difference in the means from two simulation models. - ANSWERTRUE TRUE or FALSE? The common random numbers technique intentionally induces positivecorrelation between two systems - much like a paired- confidence interval. - ANSWERTRUE CRN depends on someone's ability to manipulate the underlying pseudo-randomnumbers- e.g., use the same arrival times when simulating two competing simulated systems. Sowhoultimately controls those PRNs?? - ANSWER You do - you are powerful! TRUE or FALSE? The antithetic random numbers technique intentionally induces negativecorrelation between two runs of the same system - this allows us to better estimate themeanof the system. - ANSWER TRUE TRUE or FALSE? The control variates technique provides unbiased, low-variance 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. ONLYSTUDENTS 2025 ALL THE BEST ONLYSTUDENTS STORE 3 DO NOT COPY 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 unknown variance. Whichof thefollowing is true about a 95% confidence interval for the mean μ? - ANSWER Weare95% sure that our CI will actually contain the unknown value of μ. We are studying the waiting times arising from two queueing systems. Suppose we make4independent replications of both systems, where the systems are simulated independentlyof 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 replication are approximatelynormal, find a two-sided 95% CI for the difference in the means of the two systems. - ANSWER This is a two-sample CI problem assuming unknown and unequal variances. Wehave [-29.76, 9.76] This is sort of the same as Question 2, except we have now used common randomnumberstoinduce positive correlation between the results of the two systems. Again find a two-sided95% CI for the difference in the means of the two systems. - ANSWER This is apaired-t CI problem assuming unknown variance of the differences. [-16.5, -3.5] Suppose A and B are two identically distributed, unbiased, antithetic estimators for themeanμ of some random variable, and let C = ( A + B ) / 2. Which of the following 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 known to be equal toσ2=4. ONLYSTUDENTS 2025 ALL THE BEST ONLYSTUDENTS STORE 4 DO NOT COPY (Ya, I know that this is a crazy, unrealistic assumption, but let's go with it anyway, okeydokey?) I want to choose the best of the three populations with probability of correct selection of 95% whenever the best population's mean happens to be at least δ ⋆ = 1 larger than the second-best population's. How many observations from each population does Bechhofer's procedure N B tell me to take before I can make such a conclusion? - ANSWER Using the notation of the notes, we want to make sure to get the right answer with probability of P ⋆ = 0.95 whenever μ [ k ] − μ [ k − 1 ] ≥ δ ⋆ = 1. We simply go to NB's table with k = 3 and δ ⋆ / σ = 1 / 2 to obtain a sample size of n = 30fromeach population. In the above problem, suppose that we take the necessary observations and we comeupwiththe following sample means: X ¯ 1 = 7.6, X ¯ 2 = 11.1, and X ¯ 3 = 3.6. What do we do? - ANSWER Pick population 2 and say that we are right with probability at least 95%Suppose that we want to know which of Coke, Pepsi, and Dr. Pepper is the most popular. Wewould like to make the correct selection with probability of at least P ⋆ = 0.90 in the event that the ratio of the highest-to-second-highest preference probabilities happens to beat leastθ ⋆ = 1.4. If we use procedure M B E M, then the corresponding table in the notes (withk=3) tells us to take 126 samples (taste tests). Suppose we take those samples sequentially andafter 100 have been taken it turns out that 65 people prefer Coke, 25 love Pepsi, and10likeDr. Pepper. What to do? - ANSWER Stop the test now and declare with confidenceof at least 90% that Coke is the most-preferred. Which of the following problems might best be characterized by a finite-horizon simulation?- ANSWER Simulating the operations of a bank from 9:00 a.m. until 5:00 p.m. Let's run a simulation whose output is a sequence of daily inventory levels for a particular product. Which of the following statements is true? - ANSWER The consecutivedailyinventory levels may not be identically distributed. Suppose that X 1 , X 2 , ... is a stationary (steady-state) stochastic process with covariancefunction R k ≡ C o v ( X 1 , X 1 + k ), for k = 0 , 1 ,......We know from class that the varianceof the sample mean can be represented asV a r ( X ¯ n ) = 1 n [ R 0 + 2 ∑ k = 1 n − 1 ( 1 − k n) Rk] .We also know from class that for a simple AR(1) process, we have R k = ϕ k, k = 0 , 1 , 2, ... Compute V a r ( X ¯ n ) for an AR(1) process with n = 3 and ϕ = 0.8. - ANSWER 0.831Suppose we want to estimate the expected average waiting time for the first m= 100 customers at a bank. We make r = 4 independent replications of the system, each initializedempty and idle and co