A colleague administered a suvery and is seeking my advice. I am showing my rustiness in posing this question. A survey was sent to a universe of 445, 103 of which responded.

The population is assumed to be normally distributed. One question was binomial; 20 of the 103 responded in the affirmative.

I calculated the margin of error at +/- 7.6% on a 95% confidence level using the formula:

1. ME = z x sqrt [p(1-p))/n], where z = 1.96, and n = 103

Because the population is small and a relatively large proportion was sampled, I applied a finite population adjustment factor of 0.8714, where:

2. FPAF = sqrt (N-n)/(N-1), where N = 425 and n = 103

With this adjustment, the margin of error is +/- 6.6

This means that 95% of the time, the “true” number of yeses in the population is between 12.8% and 26%, , i.e., 86, +/- 29.

A second question generated ordinal-level data. Of the 20 yeses, a total of 24events were recorded, with the vast majority of the 20 indicating one event.

My question is, what is the most statistically valid or robust way of expressing the number of events and the margin of error at the population level? It seems to me a crude method would be to multiply the mean or modal value by the estimated number of yeses in the population, e.g., 1 x 86, +/- 29.

Is there another statistical test that is valid to apply to this data?