Margin of error, if all responses identical If I poll 10 people (with a yes/no question), and all of them respond with 'yes', should I report the rate of 'no' answers (in the greater population) is "zero plus or minus zero", or simply be confident that it is "less than one in five"?
I ask because using the "margin of error" (or "standard error of the proportion") formula sqrt[p(1-p)/n] it would appear, counter-intuitively, that the confidence interval narrows to zero (regardless of how few the samples) when the sample proportion is 0 or 1.
 A: If I flip a coin with probability of heads $=0,$ then in $n$ flips I will get $0$ heads with a standard deviation of $0$. That's OK. But the coin may instead have a non-zero probability of heads but by luck  I did not get any heads in the sample. 
The formula you give is used for a large $n$ normal approximation (CLT) to the binomial. Instead we can use an exact binomial test for small $n$. Let $q=$ proportion of "yes" voters in the population. We want to see what values of $q$ are plausible given we saw $10$ of $10$ "yes" responses. $q$ near $1$ is very likely while $q$ small, near $0$, is very unlikely. Formally, a hypothesis test:
Suppose we wanted to test:
$$H_0: q= 0.741 \text{ versus } H_a: q\gt 0.741   $$ 
This is an upper-tailed test. If we want to find the p-value corresponding to the observed result of all $10$ "yes", then we obtain $0.741^{10}=0.05$ (which is why I chose $0.741$). If we had used $0.795$ or $0.631$ we would obtain $0.795^{10}=0.10$ and $0.631^{10}=0.01.$
If we use the usual type I error $\alpha=0.05$ then we are right on the border with the stated null hypothesis and will reject the null and conclude the alternative $q\gt 0.741$ is a more plausible statement. So I would report the interval for the proportion of "yes" as $(0.741,1)$ or the range for the proportion of "no" as $(0,0.259).$ If you want to be even more conservative, we could report $(0,0.369)$ for "no" using a $1$% type I error.
A: A simple Bayesian calculation gives that if you assume before the poll that the prob $p$ of a yes is uniformly distributed on [0,1], then getting 10 yes from a randomly chosen sample of 10 gives a posterior distribution with the chance that $p>k$ as $1-k^{11}$. 
So for $k=0.8$ (corresponding to less than one in five no voters) you are 91.4% confident. 
Of course, in practice it is highly unlikely that you successfully sampled 10 people at random from the relevant population, so the calculation gets more complicated.
Any user will be deeply sceptical anyway. The same approach gives 99.6% posterior chance that $p>0.6$. But in practice I would regard 10 votes out of 10 as fairly tenuous evidence for that. If it was a political question, for example, only an amateur would poll only 10 people and any third party would doubt their competence/lack of bias.
