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Imagine that I have population of size N and I draw a sample of size n from which I compute a proportion $\hat{p}$ of a categorical variable.

And I want to get a confidence interval of 95%. So I calculate $\hat{p} \pm 1.96 \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

But the interval is too big from what I expected. Is it possible to get a smaller interval by drawing multiple samples. If yes, how to calculate the new interval.

Thank you

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    $\begingroup$ If you collect more data, you can pool them with the existing data and use the same formula. A good rule of thumb is that if your sample size were to increase by a factor of 4 the width of the confidence interval decreases by a factor of 2. It is not possible to increase precision by re-sampling within an existing dataset. $\endgroup$ – Frank Harrell Feb 11 at 12:42
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The most straightforward way is to simply combine all samples into one large sample and calculate $\hat{p}$ using this large sample (and then calculating the confidence interval using the formula you have found).

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