Central Limit Theorem Lab II (Roulette)

Summary:  Having studied individual behavior in a normal distribution, now you turn to the distribution of sample means. This lab illustrates the Central Limit Theorem for groups of 30 bets at roulette. Though the underlying population is highly non-normal, with a sample size of 30 the CLT applies fairly well.

The Population

Let’s take a highly non-normal population: the discrete probability distribution of wins and losses on $10 bets at roulette.

In US roulette, there are 38 numbers: 18 red, 18 black, and 2 green. The ball is equally likely to land on all of them. In this lab, you’ll simulate $10 bets on red: if a red number comes up, you win $10, and if a black or green number comes up, you lose $10.

Construct the discrete probability distribution at right. Look at the histogram to see why this distribution is called “highly non-normal”. Then compute the mean and standard deviation of the discrete PD:

μ = __________       σ = __________

What do μ and σ mean? Interpret them in English:

Your Sample

Now take 30 numbered slips from the supply. These are the outcomes of your 30 bets. Sort them by color, and enter the values in the table at right. Be sure to use the correct symbols for the statistics of this sample:

∑xf = __________       n = __________

x̄ = __________       s = __________

In terms of this gambling situation, write down the meaning of ∑xf:

In terms of this gambling situation, how do you interpret x̄ in English?

Sampling Distribution of the Mean

Now consider the distribution of the means of all possible samples of 30 bets from this population. What would be the mean and standard deviation of that distribution?

We can’t acually construct all possible samples, but we can use the samples gathered by your classmates to give some idea. Write down your classmates’ sample means here, as well as your own:

Now, what is the mean of those sample means? How does it compare to the mean of the population?

What is the standard deviation of those means? How does it compare to the standard deviation of the population?

You can say that the variability in the mean outcomes of 30 bets is (less, greater) than the variability in the outcomes of individual bets.

The mean you constructed in this section isn’t μ-sub-x̄, but it’s an approximation to it. If we had a few hundred samples of 30 each, instead of just a few, it would be a better approximation.

Similarly, the standard deviation of the sample means from your classmates isn’t the standard error σ_x̄, but it’s an approximation. The approximation would be better if we had lots more samples of 30. But how good is this approximation? What does the Central Limit Theorem predict for the Standard Error of the Mean (SEM or σ_x̄)?

σ_x̄ = σ / √n = __________ / √__________ = ___________

How does this compare to the standard deviation of the sample means in this class?

If we had a very big class, and plotted a histogram of everyone’s sample means, the Central Limit Theorem says it would be roughly normal, even though the original population was highly non-normal.

The Main Idea

What is the main idea you should carry with you from this lab?

The Central Limit Theorem tells you that the variability in sample means is less than the variability in the underlying population, by a factor of √n, the square root of sample size. This value, σ/√n, is called the standard error. By the 68-95-99.7 rule, a sample mean is 95% likely to be within two standard errors of the population mean, even though the individual members of the population may be quite far from the mean. This is why we can use a sample mean to approximate a population mean — but that’s a story for next week.