Sample Variability Lab (Roulette)

Summary: Having studied individual behavior in a normal distribution, now you turn to the distribution of sample means. Compared to individual bets at roulette, this lab illustrates the behavior of samples of 30 bets. You and your classmates will explore the center, spread, and shape of the distribution of sample means.

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 (see above), 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 by filling in the table. Sketch 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:

n = __________       x̅ = __________       s = __________

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

What do you see by comparing your s to your x̅? Your s to σ?

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 actually 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:

Center: What’s the mean of those sample means? How does it compare to the mean of the population?

Spread: What’s the standard deviation of those means? How does it compare to the variability of the population?

The mean you constructed in this section isn’t μ-sub-x̅, but it’s an approximation to it. If we had hundreds of 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 of the Mean (SEM or σ_x̅), but it’s an approximation. The approximation would be better if we had hundreds of samples of 30. Compute the standard error as follows:

σ_x̅ = σ / √n = __________ / √__________ = ___________

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

Shape: Your instructor will show you a histogram of the class’s sample means. Compare its shape to the histogram of the population (individual bets). 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? It is how to describe the sampling distribution of the mean:

• center: μ_x̅ = μ
• spread: σ_x̅ = σ/√n
• shape: Even if the population is non-normal, the larger the sample size, the closer the sampling distribution is to normal.

Sample means vary less than the individuals in the population do, by a factor of √n, the square root of sample size. The 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 you can use a sample mean to approximate a population mean — but that’s a story for next week.