I have a statistics question that says "What sample size would be required for the width of a $99$% confidence interval to be at most $.05$ irrespective of the value of $\hat p$?". But I'm not sure what irrespective is refering to. I tried to figure it out in my book but I can't find where exactly it explains it.
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If we are using a large sample procedure, we use $\hat{p}(1-\hat{p})$ to estimate the variance. So it plays some role in determining the width of the confidence interval.
The function $x(1-x)$ attains a maximum in the unit interval at $x=1/2$. So the most pessimistic estimate for variance, independently of the value of $\hat{p}$, is to take the variance to be $(1/2)(1-1/2)$. So we assume that the sample proportion has standard deviation $\frac{1}{2\sqrt{n}}$ for sample size $n$.
André Nicolas
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