I came across this question in my stats book in the Estimation chapter and I can quite figure it out.
The question in book is as follows:
The proportion of European men who are red-green colour-blind is 8%. How large a sample would need to be selected to be 95% certain that it contains at least this proportion of red-green colour-blind men?
I have tried equating the width of the confidence interval with the proportion that is 8%, but it doesn't seem to work. Does anybody know how to go about this question?
I think this is a something of a trick question. Since only $8\%$ of men in the population are color-blind, and we expect the sampled proportion to be normally distributed about the mean, there's really no way to be $95\%$ confident that $8\%$ of the sample will be colored blind, except by sampling the whole population, or perhaps very close to the whole population.
To take a concrete example, suppose the population size is $100$ and there are $8$ color-blind individuals. With a sample size of $90$, in order for at least $8\%$ of the sample to be color-blind, we need to sample all the color-blind men. The probability that this happens is $$\frac{\binom{92}{82}}{\binom{100}{90}}\approx.41655$$
That said, I'm not sure exactly what answer is expected.