Deck 15: Introduction to Simulation Modeling

Question 1

Which of the following statements are true?

A)A probability distribution is symmetric (around some point)if the distribution to the left of the point is a mirror image of the distribution to the right of the point.
B)We say a distribution is skewed to the right (or positively skewed)if the "longer tail" is the right tail.
C)We say a distribution is skewed to the left (or negatively skewed)if the "longer tail" is the left tail.
D)All of the above

Answer 1

D

Question 2

Which of the following statements is true regarding a simulation model?

A)It explicitly models decision-making under uncertainty
B)It explicitly incorporates uncertainty in one or more input variables
C)It provides probability distributions for all outputs,rather than expected values
D)All of these options

Answer 2

B

Question 3

We can think of the uniform distribution as:

A)the "I have no idea" distribution.
B)a skewed distribution.
C)only modeling positive values.
D)nonnegative.

Answer 3

A

Question 4

The deterministic (non-simulation)approach,using best guesses for the uncertain inputs,is:

A)better to use in complicated real world applications.
B)a good estimate of what the answer will be using a simulation approach.
C)generally not the appropriate model.
D)the preferred approach when there is correlation between input variables.

Question 5

Which of the following statements is correct regarding the graph of a discrete probability distribution?

A)It is a series of spikes.
B)The height of each spike is the probability of the corresponding value.
C)There is an empty space between adjacent spikes.
D)All of these options

Question 6

Assume that x is a random number between 0 and 1,and that the number of units expected to be sold is uniformly distributed between 300 and 500.Then,sales are given by the expression

A)300 + x
B)500 - x
C)300 + 200 x
D)500 - 200 x
E)300 + 500 x

Question 7

We sometimes use discrete distributions in place of continuous distributions:

A)because they are more accurate.
B)because they are more simple.
C)when we don't know the mean and variance of the distribution.
D)when we need to generate a histogram

Question 8

Each different set of values obtained for the uncertain quantities in a simulation model can considered to be:

A)the mean of the probability distribution.
B)a scenario.
C)a best guess.
D)all of these options.

Question 9

A continuous probability distribution is characterized by a:

A)mean and a standard deviation.
B)mean and a variance.
C)density function.
D)histogram

Question 10

Some important characteristics of probability distributions in general include the following distinctions:

A)Discrete versus continuous
B)Symmetric versus skewed
C)Bounded versus unbounded
D)Positive (or nonnegative)versus unrestricted
E)All of these options

Question 11

The RAND()function in excel models which of the following probability distributions:

A)Normal(0,1)
B)Uniform(0,1)
C)Normal(-1,1)
D)Uniform(-1,1).

Question 12

Simulation models are particularly useful for:

A)forecasting.
B)obtaining deterministic outputs.
C)evaluating constraints.
D)asking what-if questions.

Question 13

Which of the following statements are false regarding the numbers generated by the RAND function in Excel?

A)The numbers are random between 0 and 1
B)The numbers are probabilistically dependent
C)The numbers are probabilistically independent
D)The numbers are uniformly distributed between 0 and 1

Question 14

Which of the following statements are false?

A)A probability distribution is discrete if it has a finite number of possible values.
B)A probability distribution is continuous if its possible values are essentially some continuum.
C)An example of a discrete probability distribution is the amount of rain that falls during the month of June in Michigan.
D)None of these options

Question 15

A probability distribution is bounded if there are values A and B such that:

A)A and B represent the 95% confidence interval
B)A and B are the mean and standard deviation,respectively.
C)A and B are the mean and variance,respectively.
D)no value can be less than A or greater than B.

Question 16

One important special use of bounded distributions is when the only possible values are:

A)less than zero.
B)uniformly distributed around the mean.
C)skewed to the right.
D)nonnegative.

Question 17

Which of the following statements are false regarding the graph of a continuous probability distribution?

A)It is characterized by a density function,a smooth curve.
B)It is a series of spikes
C)The height of the density curve above any point is not actually a probability-that is,it is not necessarily between 0 and 1.
D)Heights above the density function indicate relative likelihoods but are not necessarily values between 0 and 1.

Question 18

Which of the following statements is (are)false regarding the numbers generated by the RAND function in Excel?

A)Approximately 10% of the numbers will be between 0.0 and 1.0
B)Approximately 20% of the numbers will be between 0.50 and 0.70
C)Approximately 40% of the numbers will be between 0.20 and 0.60
D)Approximately 60% of the numbers will be between 0.15 and 0.75
E)All of these options are false

Question 19

In order to generate random numbers in Excel from a discrete distribution with a finite number of possible values and corresponding probabilities,we can use

A)only the RAND function
B)only the VLOOKUP function
C)only the VLOOKDOWN function
D)the RAND function along with a VLOOKUP function

Question 20

The "building blocks" of all spreadsheet simulation models are:

A)deterministic inputs
B)random numbers between 0 and 1
C)decision variables
D)probability distributions

Question 21

It is simple to generate a uniformly distributed random number with a minimum and maximum other than 0 and 1.For example,the formula =150+100 RAND()generates a number uniformly distributed between

A)100 and 150
B)150 and 250
C)50 and 100
D)50 and 250

Question 22

One of the primary advantages of simulation models that they enable managers to answer what-if questions about changes in systems without actually changing the systems themselves.

Question 23

When n is reasonably large and p isn't too close to 0 or 1,the binomial distribution can be well approximated by which of the following distributions?

A)Uniform distribution
B)Normal distribution
C)Triangular distribution
D)None of these options

Question 24

If a model contains uncertain outputs,it can be very misleading to build a deterministic model by using the means of the inputs to predict an output.This is called the:

A)Law of Large Numbers.
B)Flaw of Averages
C)Law of Inevitable Disappointment
D)Central Limit Theorem

Question 25

The "random" numbers generated by the RAND function (or by any other package's random number generator)are not really random.

Question 26

Sometimes it is convenient to treat a discrete probability distribution as continuous,and vice versa.

Question 27

We typically choose between a symmetric and skewed distribution on the basis of practical modeling issues.

Question 28

A primary difference between standard spreadsheet models and simulation models is that at least one of the input variable cells in a simulation model contains random numbers.

Question 29

Suppose you run a simulation model several times with different order quantities.What can we infer about the the quantity that maximizes the output,the company's profit?

A)This quantity is the optimal order quantity.
B)This quantity might be the optimal order quantity,but we also need to consider the company's attitude toward risk.
C)This is not necessarily the optimal order quantity,because it may have produced the largest profit by luck.
D)We can't infer anything.

Question 30

A probability distribution is continuous if its possible values are essentially some continuum.

Question 31

The Excel RAND()function generates random numbers from a Normal(0,1)distribution.

Question 32

Which of the following statements are true?

A)The @RISK contains a number of functions such as RISKNORMAL and RISKDISCRETE that make it easy to generate observations from the most important probability distributions.
B)You can specify any cell or range of cells in your simulation model as output cells.When you run the simulation,@RISK automatically keeps summary measures (averages,standard deviation,percentiles,and others)from the values generated in these output cells across the replications.
C)@RISK has a special function,RISKSIMTABLE,which allows you to run the same simulation several times,using a different value of some key input variable each time.
D)All of the above

Question 33

Which of the following statements is true regarding the Triangular distribution?

A)It is a discrete distribution with a minimum,maximum and most likely value
B)It is more flexible and intuitive than the normal distribution
C)It is a symmetric distribution
D)All of these options

Question 34

Spreadsheet simulation modeling is quite similar to the other modeling applications in that it begins with input variables and then relates these with appropriate Excel formulas to produce output variables of interest.

Question 35

A probability distribution is bounded if there are values A and B such that only one possible value can be less than A or greater than B.

Question 36

If we want to model the monthly return on a stock,we might choose

A)symmetric distribution around 0
B)positively skewed distribution
C)negatively skewed distribution
D)All of these options

Question 37

Excel's built-in functions,along with the RAND function,can be used to generate random numbers from many different types of probability distributions.

Question 38

If we want to model the time it takes to serve a customer at a bank,we will probably choose

A)symmetric distribution
B)positively skewed distribution
C)negatively skewed distribution
D)All of these options

Question 39

Many companies have used simulation to determine which of several possible investment projects they should choose.This is often referred to as

A)risk analysis
B)@RISK investment
C)unbounded risk
D)None of the above

Question 40

Which of the following statements is true regarding the Normal distribution?

A)It is always the appropriate distribution in simulation modeling
B)It does not permit negative values
C)There is a 95% chance that values will be within ± 2 standard deviations of the mean
D)All of these options

Question 41

It is common in computer simulations to estimate the mean of some distribution by the average of the simulated observations.The usual practice is then to accompany this estimate with a confidence interval,which indicates the accuracy of the estimate.

Question 42

A common guideline in constructing confidence intervals for the mean is to place upper and lower bounds one standard error on either side of the average to obtain an approximate 95% confidence interval.

Question 43

Different random numbers generated by the computer are probabilistically dependent.This implies that when we generate a random number in a particular cell,it has some effect on the values of any other random numbers generated in the spreadsheet.

Question 44

The triangular distribution is sometimes used in simulation models because it is more flexible and intuitive than the normal distribution.

Question 45

The flaw of averages is the reason deterministic models can be very misleading.

Question 46

@Risk introduces uncertainty explicitly into a spreadsheet model by allowing several inputs to have probability distributions and then enabling the simulation of random values from these inputs.

Question 47

The binomial distribution is a discrete distribution that is applied to situations where n independent and identical "trials" occur,with each trial resulting in a "success" or "failure," and we want to generate the random number of successes in the n trials.

Question 48

The uniform distribution is bounded by a minimum and a maximum,and all values between these two extremes are equally likely.

Question 49

RISKSIMTABLE is a function in @Risk for running several simulations simultaneously,one for each setting of an input or decision variable.

Question 50

Analysts often plan a simulation so that the confidence interval for the mean of some important output will be sufficiently narrow.The reasoning is that narrow confidence intervals imply more precision about the estimated mean of the output variable.

Question 51

When we run simulation,the @Risk automatically keeps statistics such as averages and standard deviations,and can also create graphs such as histograms based on the values generated in the output cells in the simulation model.

Question 52

The three parameters required to specify a triangular distribution are the minimum,mean and maximum.

Question 53

When you try to find the most appropriate input probability distribution in a simulation model,you first have to choose the most appropriate family,and then you have to select the most appropriate member of that family

Question 54

If we want to model a random stock price,we should do so with an unbounded symmetric probability distribution.

Question 55

The normal distribution is often used in simulation models because it is the most common distribution in statistics and it does not allow negative values.

Question 56

The binomial distribution can be well approximated by the normal distribution when the number of trials n is sufficiently small and the probability of success p is not too close to 0 or 1.

Question 57

Data tables in spreadsheet simulations are useful for taking a "prototype" simulation and replicating its key results a desired number of times.

Question 58

A discrete distribution is useful for many situations,either when the uncertain quantity is not really continuous (the number of televisions demanded,for example)or when you want a discrete approximation to a continuous variable.

Question 59

When we maximize or minimize the value of a decision variable by running several simulations simultaneously,we have found an optimal solution to the problem and attitude toward risk becomes irrelevant.

Question 60

It is usually not too difficult to predict the shape of the output distribution from the shape(s)of the input distribution(s).

Question 61

Company experts believe the development time will be from 5 to 9 months,but they have absolutely no idea which of these will result.

Question 62

Company experts believe the development time will be from 5 to 9 months.They believe that 7 months is twice as likely as either 6 months or 8 months and that either of these latter possibilities is three times as likely as either 5 months or 9 months.

Question 63

(A)Generate the "birthdays" of 30 different people,assuming that each person has a 1/365 chance of having a given birthday (call the days of the year 1,2,3,……..,365).You can use a formula involving the INT and RAND functions to generate birthdays.
(B)Once you have generated 30 people's birthdays,you can tell whether at least two people have the same birthday using Excel's RANK function (i.e.,in the case of a tie,two numbers are given the same rank).Do you see any people with the same birthday in your sample?
(C)Obtain at least 20 samples of the 30 person group using the F9 key.What do you estimate the probability of finding two people with the same birthday in a sample of 30 people to be?

Question 64

(A)Use simulation to determine how many cars the dealer should order in August,2009 to maximize his expected profit.
(B)For the optimal order quantity,find a 95% confidence interval for the expected profit.
(C)Why is it important to develop the confidence interval in (B)?

Question 65

Correlation between two random input variables might not change the mean of an output,but it can definitely affect the variability and shape of an output disbribution.

Question 66

(A)Assume that the weight of each can in a six-pack has a 0,8 correlation with the weight of the other cans in the six-pack.What mean fill quantity (within 0.05 ounce)maximizes expected profit per sic-pack?
(B)If the weights of the cans in the six-pack are probabilistically independent,what mean fill quantity (within 0.05 ounce)will maximize expected profit per six-pack?
(C)How can you explain the difference in the answers for (A)and (B)?

Question 67

Suppose that the demand for cars is normally distributed with mean of 120 and standard deviation of 20.Use @Risk simulation add-in to determine the "best" order quantity; that is,the one that has the largest expected profit.Using the statistics and/or graphs from @Risk,discuss whether this order quantity would not be considered the "best" by the car dealer.

Question 68

A correlation matrix must always have 1's along its diagonal (because a variable is always perfectly correlated with itself)and the correlations between variables elsewhere.

Question 69

(A)The three stock returns are highly correlated.The correlation between each pair is 0.9
(B)The three stock returns are practically independent.The correlation between each pair is 0.1
(C)The first two stocks are moderately correlated.The correlation between their returns is 0.4.The third stock's return is negatively correlated with the other two.The correlation between its return and each of the first two is -0.8.
(D)Compare the portfolio distributions from @RISK for the three scenarios in (A),(B)and (C).What do you conclude?
(E)You might think of a fourth scenario,where the correlation between each pair of returns is a large negative number such as -0.80.But explain intuitively why this makes no sense.Try running a simulation with these negative correlations to see what happens.

Question 70

(A)Company experts have no idea what the distribution of the development cost is.All they can state is that "we are 90% sure it will be somewhere between $450,000 and $650,000."
(B)Company experts can still make the same two statements as in (A),but now they can also state that "we believe the distribution is symmetric and its most likely value is about $550,000."
(C)Company experts can still make the same two statements as in (A),but now they can also state that "we believe the distribution is skewed to the right,and its most likely value is about $500,000."

Question 71

Use simulation to analyze the supplier's problem.Determine how many sweatshirts he should produce to maximize the expected profit.

Question 72

If you add several normally distributed random numbers,the result is normally distributed,where the mean of the sum is the sum of the individual means,and the variance of the sum is the sum of the individual variances.This result is difficult to prove mathematically,but it is easy to demonstrate with simulation.To do so,run a simulation where you add three normally distributed random numbers,each with mean 100 and standard deviation 10.Your single output variable should be the sum of these three numbers.Verify with @RISK that the distribution of this output is approximately normal with mean 300 and variance 300 (hence,standard deviation = 17.32).

Question 73

(A)Use a simulation model to help the institute decide how many violins they must reserve with the instrument company.Consider five different possible reservation quantities: 400,500,600,700,800.Which of these quantities yields the highest total revenue,net of instrument costs?
(B)Which simulation yields the largest median total revenue?
(C)Which simulation has the most risk as measured by spread or dispersion in the data? Please state clearly what statistic you used to answer this question.
(D)Are there any simulations in which there is at least a 1 in 20 (i.e.,5%)chance of getting a negative total revenue? Briefly explain in one sentence.
(E)For each simulation what is the probability of exceeding $175,000 in total revenue (approximate these numbers as closely as possible from the data given in the above table).Please put your answer in the following table:
(F)Considering your answers for (A)through (E),please state how many instruments you think should be reserved in advance and explain why.
(G)Suppose the institute is able to negotiate with the instrument company to reduce the cost for a violin from $500 to $350.Re-run the simulation model using the same reservation quantities (but with $350 for the unused instrument cost).Has the reservation quantity that yields the highest average revenue changed? If so,please explain why this has occurred.

Question 74

Company experts believe the development time will from 5 to 9 months.They believe the probabilities of the extremes (5 and 9 months)are both 10%,and the probabilities will vary linearly from those endpoints to a most likely value at 7 months.

Question 75

A correlation matrix must always be symmetric,so that the correlations above the diagonal are a mirror image of those below it.

Question 76

Obtain another set random numbers by pressing the F9 (recalculate)key.Do your results change significantly? Do the changes match your expectations? Explain your answer.

Question 77

(A)What are the appropriate probability distributions to model the number of faculty members showing up in each lot?
(B)Given the current situation,estimate the probability that on a peak day,at least one faculty member with a sticker will be unable to find a parking space.Assume that the number who shows up at each lot is independent of the number who shows up at the other two lots.
(C)Suppose that faculty members are allowed to park in any lot.Does this help solve the problem? Why or why not?
(D)Suppose that the numbers of faculty who show up at the three lots are correlated,with each correlation equal to 0.80.Does your answer to (C)change? Why or why not?

Question 78

Use @Risk simulation add-in to analyze the sweatshirt sales.Do this for normal distributions,where we assume that the regular demand is normally distributed with mean 10,000 and standard deviation 1500,and that the demand at the reduced price is normally distributed with mean 5,000 and standard deviation 1500.

Question 79

(A)Use @Risk with 100 replications,provide a summary statistics of portfolio return; namely,minimum,maximum,mean,and standard deviation.
(B)Use your answers to (A)to estimate the probability that Mrs.Smart's portfolio's annual return will exceed 20%.
(C)Use your answers to (A)to estimate the probability that Mrs.Smart's portfolio will lose money during the course of a year.
(D)Suppose that the current price of each stock is as follows: stock 1: $16; stock 2: $18; stock 3: $20; and stock 4: $22.Mrs.Smart has just bought an option involving these four stocks.If the price of stock 1,six months from now are is $18 or more,the option enables Mrs.Smart to buy,if she desires,one share of each stock for $20 six months from now.Otherwise the option is worthless.For example,if the stock prices six months from now are: stock 1: $18; stock 2: $20; stock 3: $21; and stock 4: $24,then Mrs.Smart would exercise her option to buy stocks 3 and 4 and receive (21- 20)+ (24-20)= $5 in each cash flow.How much is this option worth if the risk-free rate is 8%?

Question 80

(A)What fraction of the random numbers are smaller than 0.5?
(B)What fraction of the time is a random number less than 0.5 followed by another random number less than 0.5?
(C)What fraction of the random numbers are larger than 0.8?
(D)What do you expect the answers to (A),(B)and (C)to be before simulating? Do the answers you provided to those questions match your expectations? Explain why or why not.
(E)Suppose your answers to (A),(B)and (C)are not close to the expected answers.What can you do to obtain answers from the simulation that are closer to the expected answers?