Question

Interpret the sample proportion,   p⁺ ,  as a sample mean.

Answer 1

The sample proportion is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. Viewed as a random variable it will be written Pˆ.P^. It has a mean μPˆμP^ and a standard deviation σPˆ.σP^. Here are formulas for their values.

Suppose random samples of size n are drawn from a population in which the proportion with a characteristic of interest is p. The mean μPˆμP^ and standard deviation σPˆσP^ of the sample proportion PˆP^ satisfy

μPˆ=p  and  σPˆ=pqn−−−√μP^=p  and  σP^=pqn

where q=1−p.