# Sample size for Wilson's algorithm

Consider a story ranking website in which the ranking is crowd sourced from the number of up votes and down votes received by a story.

The score is computed as the lower bound Wilson's algorithm.

lower bound = $(p+\frac{z^2_{\alpha/2}}{2}\pm z_{\alpha/2}\sqrt{[p(1-p)+(z^2_{\alpha/2}/4n)]/n}) / (1+\frac{z^2_{\alpha/2}}{n})$

where $p$ is the observed fraction of positive ratings, $n$ is the total ratings, and $z_{\alpha/2}$ is the $1-\alpha/2$ quantile of standard normal distribution.

Now consider two stories. One story receives votes from all the voters and the other story receives $X$ times (0<$X$<1) the votes as the first story. Wilsons algo. calculates the score $S$ the second story such that "IF" the second story had as many votes as the first story then you could say with a 95% confidence that the score is $S$.

What is the minimum $X$ for which the score remains within a $\pm i$ interval?

This question is kind of similar to Calculating the "Most Helpful" review

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