ISYE 6644 FINAL PREP EXAM WITH

ISYE 6644 FINAL PREP EXAM WITH

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ACTUAL LATEST EXAM [BRAND NEW]

What 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.

We are studying the waiting times arising from two queueing systems. Suppose we make 4 independent replications of both systems, where the systems are simulated 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 replication are approximately normal, 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. We have

[-29.76, 9.76]

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This is sort of the same as Question 2, except we have now used common random numbers to induce positive correlation between the results of the two systems.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, antithetic estimators for the mean μ 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 to σ 2 = 4. (Ya, I know that this is a crazy, unrealistic assumption, but let's go with it anyway, okey dokey?) 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 = 30 from each population.

In the above problem, suppose that we take the necessary observations and we come up with the 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. We would 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 2 / 3

probabilities happens to be at least θ ⋆ = 1.4. If we use procedure M B E M, then the corresponding table in the notes (with k = 3) tells us to take 126 samples (taste tests). Suppose we take those samples sequentially and after 100 have been taken it turns out that 65 people prefer Coke, 25 love Pepsi, and 10 like Dr. Pepper. What to do? - ANSWER- Stop the test now and declare with confidence of 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 consecutive daily inventory levels may not be identically distributed.

Suppose that X 1 , X 2 , ... is a stationary (steady-state) stochastic process with covariance function R k ≡ C o v ( X 1 , X 1 + k ), for k = 0 , 1 , .... We know from class that the variance of the sample mean can be represented asV a r ( X ¯ n ) = 1 n [ R 0 + 2 ∑ k = 1 n − 1 ( 1 − k n ) R k ] .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- 0.831

Suppose 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 initialized empty and idle and consisting of 100 waiting times. The resulting

replicate means are:

i 1 2 3 4 Z i 5.2 4.3 3.1 4.2 Find a 90% confidence interval for the mean average waiting time for the first 100 customers. - ANSWER- [3.188,5.212]

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