Quality Quiz from Professor Cleary
"B" is correct.
Click here for a more complete video explanation
Dr. Stan Deviation is exactly correct, as PQ Systems’ Quality Gamebox illustrates:
This chart shows two theoretical distributions, both symmetrical. The one on the top (green) has a single peak, and the one on the bottom (red) is the inverse of that pattern, with peaks on the left and right. When you click on the “Start” button of Gamebox, it will take 1,000 samples of the size one (n=1) from each distribution and creates a histogram of these 1,000 sample means. This is labeled as “Actual.”
Not surprisingly, both are similar to the distributions from which they were taken, but are not exactly the same, because of random variability in the 1,000 samples.
Now the fun begins.
The second time you click on “Start,” Gamebox takes 1,000 samples of the size two (n=2), averages each sample, and places these means into the distribution labeled “Actual,” below. This is of course the distribution of sample means that Emily had been explaining to Polly.
As this chart illustrates, the diagram with a single peak labeled “Actual” is tighter than the theoretical distribution. The one with two peaks changes in two ways: first, it too creates an actual distribution that is tighter than the theoretical distribution, and it goes from two peaks to three peaks.
When you click the third time, Gamebox takes 1,000 samples of the size five (n=5), averages each sample, and puts that average into actual distribution of sample means.
As you might expect by now, both distributions of sample means become tighter yet, reflecting the fact that the sample size is now 5, so the sample means will be more alike.
Finally, click the “Start” button a fourth time. Gamebox creates 1,000 samples of the size ten (n=10), averages each sample, and puts that average into the distribution of sample means:
Predictably, both distributions of sample means become tighter still. The tighter one of the two is the one that began with a single peak, rather than a double peak. Gamebox indicates this with a sigma of 6.718, and a sigma of 11.378 for the double peak data.
In my university class, students accept the fact that the single peak distribution is tighter than the theoretical distribution (and normal-shaped), but are surprised by the fact that what began as a distribution with two peaks ends with a distribution of sample means, when n=5. As the n becomes higher, the data becomes more normal looking.
This illustrates the fact that the central limit theorem is not just a theoretical concept, but one that can be proven empirically.
Finally, why did a distribution with two peaks (binomial) become a distribution of sample means with three peaks (tri-nomal)?
Tune in next month to find the answer to this perplexing question.
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