ISYE 6644 - FINAL PREP EXAM WITH CORRECT ANSWERS AND VERIFIED QUESTIONS GRADED A+ (2024 -2025) What is a po ssible goal of an indifference -zone normal means selection technique? - ANS-Find the normal population having the largest mean, especially if the largest mean is ≫ the second -largest. Name That Distribution! The number of dice tosses until a 3 comes up for the 4th time. - ANS-Negative Binomial Name That Distribution! IQs - ANS-Normal Name That Distribution! Cases in which you have limited infor mation, e.g., you only know the min, max, and "most likely" values that a random variable can take. - ANS-Triangular Find the sample variance of -3, -2, -1, 0,1,2,3 - ANS-14/3 S^2 formula If X 1 , ... , X 10 are i.i.d. Pois(6), what is the expected value of the sample variance S^2? - ANS-6 S^2 is always unbiased for the variance of X i. Thus, we have E [ S^2 ] = V a r ( X i ) = λ = 6 Suppose that estimator A has bias = 3 and variance = 12, while estimator B has bias -2 and variance = 14. Which estimator (A or B) has the lower mean squared error? - ANS-MSE = Bias^ 2 + Var, so M S E ( A ) = 9 + 12 = 21 and M S E ( B ) = 4 + 14 = 18. If X 1 = 2, X 2 = − 2, and X 3 = 0 are i.i.d. realizations from a Nor(μ , σ^2) distribution, what is the value of the maximum likelihood estimate for the variance σ^2? - ANS-σ^2 = (n -1)/n *S^2 8/3 Suppose we observe the Pois( λ ) realizationsX 1 = 5 , X 2 = 9 and X 3 = 1. What is the maximum likelihood estimate of λ? - ANS-The likelihood function is L ( λ ) Thus, ln ( L ( λ ) ) This implies that TRUE or FALSE? The Bechhofer procedure for selecting the normal populat ion 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. - ANS-True TRUE or FALSE? Sometimes a single -stage procedure like Bechhofe r's is inefficient. 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. - ANS-True For which scenarios(s) be low 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. - ANS-All three of (b), (c), and (d). Suppose that a Bernoulli selection procedure tells you to take 100 observations from each of two populations, A and B. It turns out that A gets 85 successes and B gets 46 successes. What do you think? - ANS-1) A almost certainly has a higher success 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? - ANS-
Find the most -popular pol itical candidate. Suppose that we want to know which of Coke, Pepsi and Dr.pepper 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 preferenc e probabilities happens to be at least 0*=1.4. How many people does the single -stage procedure Mbem require us to interview? - ANS-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 ca n you get confidence intervals for? Means Variances Quantiles Differences between the means of two systems - ANS-All of 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? - ANS-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 completely independent of all of the observations from the second population. - ANS-FALSE. {In fact, it's easier to distinguish between the two means if Xi is positively correlated with Yi. Think about my parallel parking example in the class notes.} TRUE or FALSE? You can use a version of independent replicatio ns to obtain confidence intervals for the difference in the means from two simulation models. - ANS-TRUE
Find the most -popular pol itical candidate. Suppose that we want to know which of Coke, Pepsi and Dr.pepper 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 preferenc e probabilities happens to be at least 0*=1.4. How many people does the single -stage procedure Mbem require us to interview? - ANS-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 ca n you get confidence intervals for? Means Variances Quantiles Differences between the means of two systems - ANS-All of 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? - ANS-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 completely independent of all of the observations from the second population. - ANS-FALSE. {In fact, it's easier to distinguish between the two means if Xi is positively correlated with Yi. Think about my parallel parking example in the class notes.} TRUE or FALSE? You can use a version of independent replicatio ns to obtain confidence intervals for the difference in the means from two simulation models. - ANS-TRUE
