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I am trying to estimate the (say 95%) confidence interval of a proportion.

For instance, I want to estimate how many students will take a particular course next year. There are N1 students in total. I asked randomly N2 students, and N3 students say yes. I estimate the proportion is N3/N2.

I tried to estimate the confidence interval as:

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Source: https://openstax.org/books/introductory-business-statistics/pages/8-3-a-confidence-interval-for-a-population-proportion

However, if p' is 0 or 1, how could I calculate the standard deviation because it will become 0, regardless of the sample size?

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  • $\begingroup$ I upgraded this question because it apparently provides a clear distinction in possible answers between a theoretical statistician and what an applied statistician, as myself, would ever consider presenting, in practice. $\endgroup$
    – AJKOER
    Commented May 17, 2020 at 19:38

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Here is an excerpt from Biostatistical Methods

One then desires a one sided confidence interval of size $1-\alpha$ of the form $(0, \hat{\pi}_u$) where the upper confidence limit satisfies the relation $\hat{\pi}_u=\pi : B(0,\pi,n)=\alpha$, the ":" meaning "such that". Solving for $\pi$ yields $$\hat{\pi}_u = 1-\alpha^{1/n}$$

When you have 0 successes in $n$ samples, your 95% confidence interval is $(0, 1-0.05^{1/n}$).

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  • $\begingroup$ Demetri: As a consulate would you really present this as your academic answer? As any non-academic personnel acquainted with the previous history of class enrollment, the school administration expects the answer is either zero or possibly, depending on other factors (like the availability of competing courses fulling an elective requirement) more than one. The real question, is how much more. $\endgroup$
    – AJKOER
    Commented May 17, 2020 at 19:12
  • $\begingroup$ @AJKOER My answer addresses the title question. If OP has design questions, then we can answer those. Furthermore, I would not consider myself a consulate in these cases. I'm not compensated in anyway other than fake internet points. $\endgroup$
    – Demetri Pananos
    Commented May 17, 2020 at 19:18
  • $\begingroup$ @AJKOER Finally, a confidence interval summarizes uncertainty at the parameter level. Your objection seems to be that the answer, whatever it may be, should be integer valued ("is either zero or possibly [...] more than one"). An integer answer is not required since the parameter of interest is always on the unit interval, so you'll have to forgive me for asking for clarification here. $\endgroup$
    – Demetri Pananos
    Commented May 17, 2020 at 19:26
  • $\begingroup$ Demetri: My answer moved from the ratio perspective to actual student counts. Again, a difference in perspective, which is crucial in the current context, as a class may have a minimum number of required students, or not, if it is a required elective. I would also claim that you have not provided the best answer as you know the context, which was provided as background. A student survey on wish list for classes, should not be viewed as equivalent to electional ballots casted for candidates (there zero votes is always zero). $\endgroup$
    – AJKOER
    Commented May 17, 2020 at 20:10
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First, you should apply the finite population correction factor (FPCF) to the standard deviation of for a small mean or proportion if your sample is in excess of 5% of the parent population. The FPCF is given by:

${FPCF =\sqrt{\frac{(N-1)}{(N-n)}}}$

Reference: See Example 7.1 (click Answer).

Now, if all N2 students said no, then N2/N1 is the proportion of students who are not intending to take the course. Construct a two-sided confidence interval (per my provided source) including the FPCF center at the sample mean.

Now, the count below the mean may (or more likely not) actually attend with probability specified by the confidence interval conditional on other factors (in essence, closer to an upper limit). For example, other available courses meeting an elective for graduation, limit on class sizes in popular courses, and required credit load that a student must carry. These constraints could, in essence, force students to select from available courses, including the one in question, for which there was no prior intent.

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