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RE:[apstat] 30 and the Central Limit Theorem
Posted:
Mar 23, 2012 7:21 PM


Bob H. and Others:
Bob writes, in part:
Cochran's rule and Sugden's work are mentioned in the second edition of Lohr's sampling text (p.44). She gives an example where the revised rule gives a minimum sample size of 193. She also says, "The 'magic number' of n=30 often cited in introductory statistics texts as a sample size that is 'sufficiently large' for the central limit theorem to apply, often does not suffice in finite popualtion sampling problems."
Whichever version of Cochran's rule you use, the minimum sample size is a function of population shape, not a constant. The samples would have to be even larger if you wanted to do onesided tests, as is common in AP.

I think to "see" the magic number 30 we should actually look at such on Page 44 of Sharon Lohr's book Sampling: Design and Analysis, which is given here (from article by R.A. Sugden and others in an issue of the Journal of the Royal Statistical Society in 2000 that was titled "Cochran's Rule for Simple Random Sampling"):
n(minimum) = 28 + 25[(SUM(y(i)ybar(universe))^3 / (NS^3))^2
for a t interval for ybar to have confidence level approx. equal to 1alpha.
This would not be difficult for APStat students to appreciate:
 If the underlying population is "perfectly" symmetric, then this becomes n(min.) = 28 + 25(0) = 28, which is the smallest value obtainable and thus gives an insight into the "30 and the CLT" concept.  Since what is being squared is a measure of skewness, students should be able to see the effect such has on choosing an appropriate sample size. [BTW, this is what Bob was referring to when he writes "the minimum sample size is a function of the population shape, not a constant."]
As an example, consider that population I have used ad nauseum, namely
2 6 8 10 10 12
but let's consider each value occurring 100 times (and so the pop. size is N=600). Thus, we'd have
n(min) = 28 + 25[(31200/(600* 34.9245))^2) = 84.
[Of course as we don't have the entire population we use the values from the sample for y(i), ybar for ybar(universe), s^3 for S^3, and n for N.]
 David Bee
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