CS 1538 – Fall 2009 – Practice Questions and Guidelines for Final Exam

There will be a variety of questions on the final exam, which will be divided into a closed book and an open book section.  You will first be given the closed book section to complete.  When you finish that section and turn it in you will be given the open book section, which you will have the remainder of the exam time to complete. 

Below is an explanation of the types of questions and an example problem or two for each type.  The solutions follow in the latter part of this document.  Closed book problems are given first, followed by open book problems.  For additional examples of closed book problems, see the Fill in the Blanks + True / False and the Short Answer practice questions from the midterm exam (as well as those sections on the midterm exam itself).

I strongly recommend giving the problems a good try before looking at the solutions.  Note that the problems shown are only example problems and are not necessarily representative in number of the questions that will actually be on the exam.  Thoroughly read over the Review Sheet to see the material that you need to know for this exam.

CLOSED BOOK EXAMPLE PROBLEMS (see midterm practice questions and midterm exam for example problems from first part of the term)

Fill in the Blanks and True/False – these problems will deal mainly with definitions and possibly simple computations.  For false questions you will be expected to explain WHY they are false.

1)    If we determine a function F-1 that maps values from U[0,1) into some other distribution, we are using the ___________________________________________________ to generate that distribution.

 

2)    A b, or Type II error is the probability that _________________________________________________

 

3)    For a given a confidence level, we can REDUCE the b error value by ______________________________ _______________________________________________.

 

4)    The process of testing the implementation of a simulation model to ensure that it has been programmed correctly and debugged is called model __________________________________.

 

5)    If an output value of interest, Y, is averaged over 10 runs of a simulation, we call the result, Y_bar, the _________________________________________ of the actual mean value.

 

6)    (T/F) Interarrival times for a simulation are typically represented by decision variables.

 

7)    The t-distribution approaches a standard normal distribution as the degrees of freedom approach ¥.

 

8)    Comparing alternative system designs using common random numbers will likely have _______________________________ [higher, lower] sample variances than comparing using independent random numbers.

 

Short Answer – these problems will typically include definitions, comparisons and explanations.  There may be some calculations, but most of the calculations will be in the Problem Solving below. (see midterm practice questions and midterm exam for example problems from first part of the term)

1)    For pseudo-random number generators to be useful, they should produce sequences over U[0,1) that are uniform and independent.  Informally state what each of these properties means.  Give and justify an example of a sequence that is uniform over U[0,1) but not independent, and a sequence that is independent but not uniform over U[0,1).

2)    Consider the notion of Face Validity of a simulation model.  Explain what this is and give an example of a model that fails the Face Validity test.

3)    Consider the following result of a confidence interval test:

Y_bar – ta/2,f (s_bar(Y_bar)) ≤ µ ≤ Y_bar + ta/2,f (s_bar(Y_bar))

Explain in words (and pictures, if it will help) what this means.  Be detailed and specific, identifying each term and what the overall expression means.

OPEN BOOK EXAMPLE PROBLEMS

Problem Solving – these problems will be similar in nature to problems done in lecture and assigned in Written Assignment 3 and Written Assignment 4.  They will involve use of formulas / equations / procedures that we discussed in lecture as well as some problem-analyzing and problem-solving skills.   Be sure to bring your text and other printed documents with you to prepare you for these problems.  I will also allow you to view the online notes and to use Excel on the lab computers.  I have only included three problems below.  Refer to your homework problems and textbook / in-class examples for additional problems.

1)    A proposed random number generator produced the following sequence of values over U[0,1):

0.3   0.8  0.1  0.9  0.5

Using the Kolmogorov-Smirnov test, determine whether or not this sequence is uniform over U[0,1) with a level of significance of a = 0.05.  Be specific and show all work in the process.

2)    Develop a generator for the random variable whose cdf is

F(x) =  x/3                               for 0 ≤ x ≤ 2

            2/3 + (x-2)/24              for 2 < x ≤ 10

3)    Problem 10.2 from the Banks text:  System data for the job shop of Exercise 1 (no need to know Exercise 1 to answer this problem, however) revealed that the average time spent by a job in the shop was approximately 4 working days.  The model made the following predictions on seven independent replications, for average time spent in the shop:

3.70   4.21  4.35  4.13  3.83  4.32  4.05

Is the model output consistent with system behavior?  Conduct a statistical test using a level of significance a = 0.01.

Miscellaneous – problems here may include simple traces, writing small code segments or other problems that don't fit into the other 3 categories.

Consider the set of interarrival times below.  Draw a histogram of these times (choosing a reasonable interval for the bins) and make a preliminary guess on the distribution based on that histogram.

62, 47, 71, 45, 65, 43, 58, 35, 55, 31, 51, 25, 52, 56, 49

 

 

 

 

 

 

 

SOLUTIONS

Don't look below until you have given all of the problems a GOOD try!



 

 

 

 

 

 

CLOSED BOOK PROBLEMS

Fill in the Blanks and True/False – these problems will deal mainly with definitions and possibly simple computations.  For false questions you will be expected to explain WHY they are false.

1)    If we determine a function F-1 that maps values from U[0,1) into some other distribution, we are using the _________INVERSE TRANSFORM TECHNIQUE_______________ to generate that distribution.

 

2)    A b, or Type II error is the probability that ___H0 will NOT be rejected when it should be_____

 

3)    For a given a confidence level, we can REDUCE the b error value by ____increasing the number of runs___.

 

4)    The process of testing the implementation of a simulation model to ensure that it has been programmed correctly and debugged is called model ________verification______________.

 

5)    If an output value of interest, Y, is averaged over 10 runs of a simulation, we call the result, Y_bar, the _________point estimate__________________ of the actual mean value.

 

6)    (T/F) Interarrival times for a simulation are typically represented by decision variables.  FALSE – interarrival times are typically uncontrollable variables, since they are determined external to the system.

 

7)    The t-distribution approaches a standard normal distribution as the degrees of freedom approach ¥.  TRUE

 

8)    Comparing alternative system designs using common random numbers will likely have ________lower__________________ [higher, lower] sample variances than comparing using independent random numbers.

Short Answer – these problems will typically include definitions, comparisons and explanations.  There may be some calculations, but most of the calculations will be in the Problem Solving below.

1)    For pseudo-random number generators to be useful, they should produce sequences over U[0,1) that are uniform and independent.  Informally state what each of these properties means.  Give and justify an example of a sequence that is uniform over U[0,1) but not independent, and a sequence that appears to be independent but not uniform over U[0,1).

Answer:  If the values are uniform, then the probability that a value will be anywhere within the range of values is equally likely.  If they are independent, then previous values do not (appear to) affect future values and there are no discernable patterns in the values.

Sequence that is uniform but not independent:  0.05, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75, 0.85, 0.95.  This sequence is uniform because it covers the range completely (it would easily pass a Komolgorov-Smirnov test for uniformity) but is not independent because each Xi+1 = Xi + 0.1.

Sequence that is independent but not uniform: 0.25, 0.12, 0.38, 0.53, 0.32, 0.44, 0.39, 0.04, 0.26, 0.19.  This sequence appears to be independent (there are no obvious correlations) but is clearly not uniform, since the maximum value is 0.53 and the average will be much less than the expected average of ~0.5.

2)    Consider the notion of Face Validity of a simulation model.  Explain what this is and give an example of a model that fails the Face Validity test.

Answer: See slide 238 in notes and p. 362 of text for definitions.  Consider as an example the Panera simulation.  If the input distribution from the doors is the same during all hours of operation, the model would not be valid since it is obvious to informed (and even uninformed) parties that this distribution should be different at different times of day.

3)    Consider the following result of a confidence interval test:

Y_bar – ta/2,f (s_bar(Y_bar)) ≤ µ ≤ Y_bar + ta/2,f (s_bar(Y_bar))

Explain in words (and pictures, if it will help) what this means.  Be detailed and specific, identifying each term and what the overall expression means.

 

Answer: The value µ is the "actual" mean value of the output variable of interest, Y.  Y_bar is the point estimate of this mean and s_bar is the standard error of the point estimator.  ta/2,f  is the value in the standard t-distribution with f degrees of freedom.  The product, ta/2,f (s_bar(Y_bar)), is the half-length of the confidence interval.  Overall, the expression is saying that the actual mean, µ , will be within a half-length of the point estimate with a  probability 1-a.

 

OPEN BOOK PROBLEMS

Problem Solving – these problems will be similar in nature to problems done in lecture and assigned in Written Assignment 3 and Written Assignment 4.  They will involve use of formulas / equations / procedures that we discussed in lecture as well as some problem-analyzing and problem-solving skills.   Be sure to bring your text with you to prepare you for these problems.  I will also allow you to view the online notes and to use Excel on the lab computers.  I have only included three problems below.  Refer to your homework problems and textbook / in-class examples for additional problems.

1)    A proposed random number generator produced the following sequence of values over U[0,1):

0.3  0.8  0.1  0.9  0.5

Using the Kolmogorov-Smirnov test, determine whether or not this sequence is uniform over U[0,1) with a level of significance of a = 0.05.  Be specific and show all work in the process.

Answer: First we sort the values in increasing order, yielding: 0.1, 0.3, 0.5, 0.7, 0.9.  Next we find the D+ and D- values, getting the maximum positive and negative differences of our sequence from the uniform function F(x) = x.  Setting things up, we get:

Ri:                    0.1       0.3       0.5       0.8       0.9
i/N:                  0.2       0.4       0.6       0.8       1.0
i/N-Ri:             0.1       0.1       0.1       0.0       0.1       D+ = max of these = 0.1
Ri-(i-1)/N:       0.1       0.1       0.1       0.2       0.1       D- = max fo these = 0.2

D = max(D+, D-) = 0.2

Looking up the critical value from Table A.8 for a = 0.05 with n = 5, we get 0.565.  Since D is less than this value, our values pass the K-S test.
 

2)    Develop a generator for the random variable whose cdf is

F(x) =  x/3                               for 0 ≤ x ≤ 2

            2/3 + (x-2)/24              for 2 < x ≤ 10

Answer: Consider the subrange 0 ≤ x ≤ 2.  Based on the cdf, let R = x/3.  Plugging in the range endpoints for x, we get 0 ≤ R ≤ 2/3.  Solving for x, we get x = 3R.
Consider the subrange 2 < x ≤ 10.  Based on the cdf, let R = 2/3 + (x-2)/24.  Plugging in the range endpoints for x, we get 2/3 < R ≤ 1.  Solving for x, we get x = 2 + 24(R – 2/3) = 24R – 14.  Now we can complete our generator as:

X =      3R                   0 ≤ R ≤ 2/3
            24R – 14         2/3 <  R ≤ 1

3)    Problem 10.2 from the Banks text:  System data for the job shop of Exercise 1 (no need to know Exercise 1 to answer this problem, however) revealed that the average time spent by a job in the shop was approximately 4 working days.  The model made the following predictions on seven independent replications, for average time spent in the shop:

3.70     4.21     4.35     4.13     3.83     4.32     4.05

Is the model output consistent with system behavior?  Conduct a statistical test using a level of significance a = 0.01.

Answer:  We assume µ = 4.  Conduct a t-test using the 7 values against the mean (using the TTEST function in EXCEL), revealing an answer of 0.38398.  Using the TINV function on this value with 6 degrees of freedom, we get the t-value = 0.913476.   Since our a level of significance is 0.01, we use the TINV function on this value with 6 degrees of freedom to get the critical value in this case = 3.7074.  Since our t-value is well under the critical value, we do not reject the null hypothesis.

Note that this problem could also be solved using the formulas in the text, in the following fashion:

Y_bar = 4.08  Sy = 0.2441 (using the formulas on p. 369 of the text)
From this we can use Equation 10.2 to get t0 = (Y_bar – µ)/(Sy/n1/2) = (4.084 – 4)/(0.2441/71/2) = 0.91.  The rest of the answer is the same as above.

Miscellaneous – problems here may include simple traces, writing small code segments or other problems that don't fit into the other 3 categories.

Consider the set of interarrival times below.  Draw a histogram of these times (choosing a reasonable interval for the bins) and make a preliminary guess on the distribution based on that histogram.

62, 47, 71, 45, 65, 43, 58, 35, 55, 31, 51, 25, 52, 56, 49

Answer:

Try intervals of 10 as follows:

20-29   30-39   40-49   50-59   60-69   70-79
-------   ------    ------    ------    ------    ------
    1         2          4          5          2          1

Without analysis, the result appears to be normal.  Naturally, analysis is needed to verify this assertion.