Statistics, Parameters, and the Central Limit Theorem

[Bacle]

Hi, everyone:
I am teaching an intro. stats course, and I want to find a convincing explanation of how we can "reasonably" estimate a population parameter by taking random samples. Given that the course is introductory, I cannot do a proof of the CLT.


Specifically, what has seemed difficult in previous years for many students to accept, is that one can estimate a parameter (with any degree of confidence)from a population of around 310 million (current U.S pop.) by taking a random sample of size, say n=10,000 or less.

AFAIK, the Central Limit Theorem is used to explain the representability of the larger population in a sample, from the fact that, informally (please correct me if I am wrong) biases, or deviations from the average cancel each other out, so that the aggregate
deviates less from the mean, i.e., the standard deviation decreases as the sample size increases..

Would someone please comment on the accuracy of this statement and/or offer refs. about it.?

Thanks in Advance.

 

[bpet]

Hi, everyone:
I am teaching an intro. stats course, and I want to find a convincing explanation of how we can "reasonably" estimate a population parameter by taking random samples. Given that the course is introductory, I cannot do a proof of the CLT.


Specifically, what has seemed difficult in previous years for many students to accept, is that one can estimate a parameter (with any degree of confidence)from a population of around 310 million (current U.S pop.) by taking a random sample of size, say n=10,000 or less.

AFAIK, the Central Limit Theorem is used to explain the representability of the larger population in a sample, from the fact that, informally (please correct me if I am wrong) biases, or deviations from the average cancel each other out, so that the aggregate
deviates less from the mean, i.e., the standard deviation decreases as the sample size increases..

Would someone please comment on the accuracy of this statement and/or offer refs. about it.?

Thanks in Advance.

Sounds kind of on the right track - if they can understand that the variance of an average is (1/n) times the variance of the population then that should be enough to see why a sample size 10,000 is "big enough" no matter how big the whole population is, provided the sample is truly random. CLT just makes explicit the additional conditions for the limit to converge.

Good luck with it!

 

[statdad]

it's very easy to set up some small populations in Minitab (or most other software), calculate the mean and sd for the population, then demonstrate all possible samples of size 2 (or 3, if the original population isn't too large) and show how the mean of that population of samples is unchanged but the sd has decreased. graphs of the original population and the population of sample means will help as well. (this is even possible if (insert shudder of horror here) you are using Excel as a teaching aid.