Proportion Estimation

Proportion estimation is another common task for sample statistics.

We have sample proportion

${\hat {p}}={\frac {y}{n}}$

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

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 \mu_\hat{p} = p, \sigma_\hat{p} = \sqrt{\frac{p (1 - p)}{n}} }

and standard error

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

Assumptions

We assume that

• A random sample was taken
• 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 \geq 5} 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 - y \geq 5}
  • rooted in normal approximation of binomial

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} = \frac{y + 2}{n + 4} }

with standard error

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 SE(\widetilde{p}) = \sqrt{\frac{\widetilde{p} (1 - \widetilde{p})}{n + 4}} }

Remember that the confidence interval is ca

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}} is slightly skewed towards 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 0.5} , but results in better CIs 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} . I don't know why.

Confidence Interval

We use normal distribution since 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} is bounded between 0 and 1, and we don't have extra error from extra parameters such as multiple sample mean.

Remember that the confidence interval is just mean plus-or-minus error margin, and the error margin is just the z score multiplied by standard error (since we are using normal distribution).

Notaby, it is possible to have a bound above 1 or below 0. This usually happens when the point estimate is close to 0 or 1. In this case, instead of listing the impossible bounds, we report that they have been cut off.