Monte Carlo studies are useful in situations where the researcher feels that a certain probability distribution is a realistic model for the outcomes of the experiment.
The advantage of a Monte Carlo study is that it is much less expensive than collecting real data. A sample of size 10,000 actual measurements would be expensive and would take a long time to collect and validate, but 10,000 random numbers can be generated in less than a second with a computer at very little cost.
One disadvantage of this kind of study is that the result is only as good as the assumption that the outcomes follow a certain probability distribution.
In this problem we will assume that a certain population has an average height of 67 inches (5 feet, 7 inches), and that heights follow a bell curve (normal) distribution with a standard deviation of 3.5 inches.
1) Generate a simulated sample of size 1,000 persons from this population.
When you select the Normal distribution from the pulldown menu, in the dialog box enter the following parameters:
Use MINITAB to draw a histogram of this data.
Now generate a second sample of 1,000 persons, but this time assume the standard deviation is 2: When you select the Normal distribution from the pulldown menu, in the dialog box enter the following parameters:
2) Use the histogram to answer the following questions:
In this problem we will assume that baseball teams R and Y play each other 10 times during the regular season. We will assume for the purpose of the experiment that the probability that team R wins is the same every time they play, and that the outcome of any one game has no effect on the outcome of the other games they play. Under these conditions, the number of games won by team R follows a Binomial probability distribution.
Suppose the probablilty that team R wins is 0.5 for each game, that is, the teams are evenly matched.
Use MINITAB to generate a sample representing 1,000 seasons with 10 games between R and Y each season.
When you select the Binomial distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of a sample of 1,000 seasons with equally matched teams playing 10 games.
Now modify the experiment so that team R has a 60% chance of winning each game:
Generate a histogram using MINITAB of a sample of 1,000 seasons teams playing 10 games where R has a 60% chance of winning each game.
Manufacturers of incandescent light bulbs generally state the average time to burnout on the package. We will assume for the purpose of the experiment that the time to burnout of an individual bulb follows the Exponential distribution (a common choice for this type of analysis).
Use MINITAB to generate a sample representing 1,000 light bulbs with an average life of 900 hours.
When you select the Exponential distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of the simulated burnout times of the 1,000 bulbs in the sample.
Suppose an extended life bulb is available with an average life of 1,500 hours. Use MINITAB to generate a sample representing 1,000 light bulbs of this type.
When you select the Exponential distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of the simulated burnout times of the 1,000 bulbs in the second sample.
A cell phone service needs to plan the capacity of its network in a certain area. We will assume for the purpose of the experiment that the average number of calls placed from the area per minute is 50, and that the number of calls in a given minute follows the Poisson distribution (a common choice for this type of analysis).
Use MINITAB to generate a sample representing 10,800 minutes of call history (equivalent to twenty nine-hour business days, or one month of business hours).
When you select the Poisson distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of the simulated number of calls in each of the 10,800 minutes in the sample.
Now suppose the call volume actually averages 100 calls per minute. Use MINITAB to generate a sample representing 10,800 minutes of call history (equivalent to twenty nine-hour business days, or one month of business hours).
When you select the Poisson distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of the simulated number of calls in each of the 10,800 minutes in the sample.
A commonly used technique for estimating the size of a biological population is the capture-recapture method:
If the tagging has no effect on the probability that an individual is recaptured or dies, and enough time elapses between samples that the tagged and untagged individuals become thoroughly mixed, the number of tagged individuals in the second sample follows the Hypergeometric distribution.
Use MINITAB to generate a sample representing 1,000 capture-recapture experiments in which 300 lobsters in a certain area are trapped, tagged, and released. A sample of 100 lobsters is then captured and the number of tagged individuals is recorded.
We will first suppose that there are 5,000 lobsters in the area.
When you select the Hypergeometric distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of the simulated number of tagged lobsters in the second sample.
Now simulate another 1,000 capture-recapture experiments, but this time suppose that there are actually 15,000 lobsters in the area.
When you select the Hypergeo distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of the simulated number of tagged lobsters in the second sample, and use the two histograms to answer the following questions:
In the baseball World Series, two teams R and C play until one team has won four games, at which point the series ends. Under the assumption that the probability that team R wins is the same in each game, and the outcome of one game has no effect on the outcome of the others (rather questionable assumptions, but we will make them for the sake of this experiment), if the teams were to simply play until team R has won four games, the number of games required would be described by the Negative Binomial distribution.
Use MINITAB to generate a sample representing 1,000 series where the teams play until team R has won 4 games, regardless of how many games that takes. In terms of the World Series, you would interpret any series of 7 games or less to be a World Series won by R, and anything longer than 7 games to be a World Series won by C.
When you select the Negative Binomial distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a histogram using MINITAB of the simulated number of games needed for R to win 4 games if the teams are evenly matched.
Now suppose that R is actually the better team, and R has a 60% chance of winning each game.
When you select the Negative Binomial distribution from the pulldown menu, in the dialog box enter the following parameters:
Generate a second histogram of the simulated number of games needed for R to win 4 games if R has a 60% chance of winning each game.
| Name | Problem Assigned |
| Adam_S | 3 |
| Amanda_S | 3 |
| Andrew_B | 3 |
| Caitlin_C | 6 |
| Caitlin_R | 5 |
| Casey_G | 4 |
| Christina_S | 6 |
| Christina_W | 2 |
| Erin_G | 4 |
| Gregory_S | 2 |
| Ian_D | 5 |
| Janelle_D | 5 |
| Jessie_M | 5 |
| Justin_S | 3 |
| Khalid_A | 3 |
| Lauren_B | 5 |
| Lauren_S | 4 |
| Margaux_F | 2 |
| Nicholas_P | 5 |
| Patrick_M | 4 |
| Patrick_O | 3 |
| Rebecca_W | 6 |
| Sanjay_M | 2 |
| Terrell_C | 4 |
| Thomas_S | 4 |
See the Histogram HowTo page for help with MINITAB histograms.