Monte Carlo Simulation

Explanation

The goal of INFERENTIAL STATISTICAL analysis is to make PROBABILITY statements about a POPULATION PARAMETER, θ, from a statistic, ^θ, calculated from a sample ofdatafroma POPULATION. This is problematic because the calculated statistics from any two RANDOM SAMPLES from the same population will almost never be equal. For example, suppose you want to know the average IQ of the 20,000 students at your university (θ), but you do not have the time or resources to test them all. Instead, you draw a random sample of 200 and calculate the mean IQ score (^θ) of these sampled students to be, say, 123. What does that tell you about the average IQ of the entire student body? This is not obvious, because if you drew another random sample of 200 from those same 20,000 students, their mean IQ might be 117. A third random sample might produce a mean of 132, and so on, through the very large number of potential random samples that could be drawn from this student body. So how can you move from having solid information about the sample statistic (^θ) to being able to make at least a probability statement about the population parameter (θ)?

At the heart of making this inference from sample to population is the SAMPLING DISTRIBUTION. A statistic's sampling distribution is the range of values it could take on in a random sample from a given population and the probabilities associated with those values. So the mean IQ of a sample of 200 of our 20,000 students might be as low as, say, 89 or as high as 203, but there would [Page 663]probably be a much greater likelihood that the mean IQ of such a sample would be between 120 and 130. If we had some information about the structure of the sample mean's sampling distribution, we might be able to make a probability statement about the population mean, given a particular sample.

In the standard parametric inferential statistics that social scientists learn in graduate school (with the ubiquitous t-tests and p values), we get information about a statistic's sampling distribution from mathematical analysis. For example, the CENTRAL LIMIT THEOREM gives us good reason to believe that the sampling distribution of the mean IQ of our random sample of 200 students is distributed normally, with an expected value of the population mean and a STANDARD DEVIATION of approximately the standard deviation of the IQ in the population divided by the square root of 200. However, there are situations when either no such analytical distributional theory exists about a statistic or the assumptions needed to apply parametric statistical theory do not hold. In these cases, one can use Monte Carlo simulation to estimate the sampling distribution of a statistic empirically.

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