Proportion Estimation

Proportion estimation is another common task for sample statistics.

We have sample proportion

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p} = \frac{y}{n} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is the number of subjects in the sample with a particular trait, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is the sample size.

We have

$\mu _{\hat {p}}=p,\sigma _{\hat {p}}={\sqrt {\frac {p(1-p)}{n}}}$

and standard error

$SE={\sqrt {\frac {{\hat {p}}(1-{\hat {p}})}{n}}}$

Wilson-Adjusted CI for p

Correcting the sample proportion narrows the confidence interval. We do this with the Wilson-Adjusted estimate for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \widetilde{p} }