ISYE 6644 COMPREHENSIVE UPDATED EXAM
QUESTIONS AND ANSWERS SURE A+
✔✔TRUE or FALSE? You can use a version of independent replications to obtain
confidence intervals for the difference in the means from two simulation models. -
✔✔TRUE
✔✔TRUE or FALSE? The common random numbers technique intentionally induces
positive correlation between two systems - much like a paired-𝑡confidence interval. -
✔✔TRUE
✔✔CRN depends on someone's ability to manipulate the underlying pseudo-random
numbers - e.g., use the same arrival times when simulating two competing simulated
systems. So who ultimately controls those PRNs?? - ✔✔You do - you are powerful!
✔✔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. - ✔✔TRUE
✔✔TRUE or FALSE? The control variates technique provides unbiased, low-variance
estimates using a method reminiscent of regression. - ✔✔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. - ✔✔All of the above.
, ✔✔Suppose we are dealing with i.i.d. normal observations with unknown variance.
Which of the following is true about a 95% confidence interval for the mean μ? - ✔✔We
are 95% 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
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. -
✔✔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 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. - ✔✔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? - ✔✔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? - ✔✔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 = 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? - ✔✔Pick population 2 and say that we are right with probability at least
95%
QUESTIONS AND ANSWERS SURE A+
✔✔TRUE or FALSE? You can use a version of independent replications to obtain
confidence intervals for the difference in the means from two simulation models. -
✔✔TRUE
✔✔TRUE or FALSE? The common random numbers technique intentionally induces
positive correlation between two systems - much like a paired-𝑡confidence interval. -
✔✔TRUE
✔✔CRN depends on someone's ability to manipulate the underlying pseudo-random
numbers - e.g., use the same arrival times when simulating two competing simulated
systems. So who ultimately controls those PRNs?? - ✔✔You do - you are powerful!
✔✔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. - ✔✔TRUE
✔✔TRUE or FALSE? The control variates technique provides unbiased, low-variance
estimates using a method reminiscent of regression. - ✔✔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. - ✔✔All of the above.
, ✔✔Suppose we are dealing with i.i.d. normal observations with unknown variance.
Which of the following is true about a 95% confidence interval for the mean μ? - ✔✔We
are 95% 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
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. -
✔✔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 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. - ✔✔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? - ✔✔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? - ✔✔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 = 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? - ✔✔Pick population 2 and say that we are right with probability at least
95%
