10.1 Population Proportion and the Sample Proportion

Recall that the population mean $\mu = \frac{\sum x_i}{N}$ is a population parameter used to describe the population, where N is the population size (number of individuals in the population). The population proportion

$p = \frac{\text{# of individuals having a certain attribute}}{\text{population size}} = \frac{\text{# of successes}}{N}$

is another parameter used to describe the population. For example, the proportion of female students at MacEwan is defined as

$p = \frac{\text{# of female students at MacEwan}}{\text{total number of students at MacEwan}} = \frac{\text{# of successes}}{N}.$

In this instance, picking a female student is regarded as a success.

Just as the sample mean $\bar{x} = \frac{\sum x_i}{n}$ is used to estimate the population mean $\mu$, the sample proportion $\hat{p}$ is used to estimate the population proportion p, where

$\hat{p} = \frac{\text{# of individuals having a certain attribute in the sample}}{\text{sample size}} = \frac{\text{# of successes in the sample}}{n}.$

Here are several examples: