Question

95% confidence; n=320, x=60. Find the margin of error E that corresponds to to the given statistics and confidence level.

Answer 1

So find the z-score corresponding to a left or right tail of \(0.025\), this will be your critical value \(z^*\) up to a sign change.

Since we're dealing with a binomial setting, we have sample mean \(\hat{p}=\frac{x}n=\frac{60}{320}=\frac3{16}\) and an estimate for the standard deviation \(\sigma\approx\sqrt{\hat{p}(1-\hat{p})}=\sqrt{\frac3{16}\times\frac{13}{16}}=\frac{\sqrt{39}}{16}\) -- and thus our sample error is \(\mathbf{S.E.}_{\hat{p}}=\frac\sigma{\sqrt{n}}=\frac{\sqrt{39}}{16\sqrt{320}}\).

Our margin of error is then merely \(z^*\mathbf{S.E.}_{\hat{p}}\).