Margin of error--again

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Margin of error--again
Message posted by Jim on October 11, 2000 at 12:00 AM (ET)

Thanks to those who answered my previous query (my particular thanks to Nancy Diehl), I learned that the margin of error for a 95% confidence rating in a questionnaire of a single yes and no question is given by the equation:

1.96*[SQRT(p(1-p)/n)]

where p is the ratio of the #yes answers to the questionnaire to the #no answers, and n is the number of questionnaires returned. I assume this presumes an infinite sample size. In my case the same size is fairly small. I expect to receive about 40 questionnaires returned from a sample size of about 300. So my question is: How is the above formula modified to include the finite same size?

Thanks again.

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Re: Margin of error--again
Message posted by nancy diehl on October 11, 2000 at 12:00 AM (ET)

The problem I see is that your sample size is not exactly a large one. You are expecting to receive 40 answers back. So to be clear here, let's say that out of the 40, 15 reply yes,
so your p, proportion, is 15/40 = .37. Then proceed with the formula stated earlier where your n is 40. The only concern I have with using this is that the sample I believe would be
considered small, not large. The binomial distribution can be shown to approximate the normal when the sample size is large. Not much is offered in introductory statistics if the
sample size is small. However, I have a few suggestions. The variability will not be larger than when p is .5. So rather than use your real proportion of p=.37, use .5 instead. That
would offer a worse case scenario. My other suggestion is to find resources how to calculate the margin of error using the Poisson distribution as anapproximation to the binomial. Lastly,
there is something called "Exact Intervals" from a book by Feller (Wiley publication, 1968), Ch 10, equatn 10.7 that could also provide you an estimate of margin of error when your sample
is small.

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