This is probably a silly question, but I do not have a ready answer for it, so I thought I'd get some opinions on it.
Let Y ~ Normal($ {\bf \unicode[Times]{x3Bc}} $, $ {\bf \unicode[Times]{x3C3}}^2 $). We take a large random sample of 1000 observations from Y. We want to make an inference about the value of $ {\bf \unicode[Times]{x3C3}}^2 $. The standard approach is to make use of the result that the standardized sample variance below has a chi-square distribution with n-1 degrees of freedom.
What if instead, we did the following. We use the fact that for large n, a binomial distribution is well approximated by the Normal distribution, and use the sample at hand to infer the value of the probability parameter p of the approximating Binomial distribution. In other words, we know that Y is Normally distributed, but (for large enough n) there must be a Binomial distribution that approximates this Normal distribution very well. Usually this fact is used in the other direction: Normal approximation to the Binomial instead of the Binomial approximation to the Normal, but nothing says we cannot use this fact in the other direction. Finally, knowing that the variance of the binomial is np(1-p), using basic algebra we have ourselves an interval estimate of the population variance, without relying on the chi-square distribution.
My questions are:
- Does this methodology not make sense because I made a logical error somewhere or misstated some fact?
- Assuming this methodology makes sense, is it not used because it is more tedious than using the chi-squared distribution or because it produces inferior estimates of the population variance? If it is more tedious, which step is?