Quality Quiz from Professor Cleary
Congratulations:
"Yes" is correct.
Click here for a more complete video explanation
Answer: Yes: For once, Polly has taken time to think things through, and has come up with an accurate analysis.
Last month we observed that by taking sample sizes of two (n=2) from a binomial population, averaging these samples and forming a new distribution, we would see a trinomial distribution.
Clicking Start on the Quality Gamebox when the sample size is two (n=2), the program takes 1,000 samples of size two, averages each of these samples, and forms a new distribution with these 1,000 sample means. Although Gamebox creates this new distribution quickly, it is useful to examine the theory behind this process.
Taking a sample size of two (n=2) from any binomial distribution can lead to the following, where the observation could be from the left side or the right side.
Assuming that the first sample is from the left side, the second sample could be from the left side or right side as well.
Assuming that the first sample is from the right side, the second sample could also be from the left side or the right side.
With a sample size of two, four outcomes are possible. Since each of these events is equally likely, the probabilities are:
Left left 25%
Left right 25%
Right left 25%
Right right 25%
This explains how a tri-modal distribution ensues. The hump on the left of the distribution is the Left/Left (25%); the hump in the middle is Left/Right (OR Right/Left) (50%), and the hump on the right represents Right/Right (25%).
This background provides a good quiz for your colleagues: “Why does a binomial distribution become a tri-modal distribution when you take many samples of the size two (n=2) and form a new distribution of the means of these samples?”
While this may be interesting, here’s the punch line: When you hit the Start button on Gamebox, it will again take 1,000 samples, but this time it will be sample sizes of five (n=5). It will then take these 1,000 samples and average them, then put these averages into a new distribution. These distributions show that the single peaked and the binomial both become normal when you take 1,000 samples of the size of five (n=5) and plot the means of these samples.
This of course reinforces the third part of the central limit theorem that, “the shape of the distribution of sample means will be normal no matter what the shape of the population.”
Next month we will take this theory and find out how to build confidence intervals from a single sample.
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