10.1 Population Proportion and the Sample Proportion
Recall that the population mean [latex]\mu = \frac{\sum x_i}{N}[/latex] is a population parameter used to describe the population, where N is the population size (number of individuals in the population). The population proportion
[latex]p = \frac{\text{# of individuals having a certain attribute}}{\text{population size}} = \frac{\text{# of successes}}{N}[/latex]
is another parameter used to describe the population. For example, the proportion of female students at MacEwan is defined as
[latex]p = \frac{\text{# of female students at MacEwan}}{\text{total number of students at MacEwan}} = \frac{\text{# of successes}}{N}.[/latex]
In this instance, picking a female student is regarded as a success.
Just as the sample mean [latex]\bar{x} = \frac{\sum x_i}{n}[/latex] is used to estimate the population mean [latex]\mu[/latex], the sample proportion [latex]\hat{p}[/latex] is used to estimate the population proportion p, where
[latex]\hat{p} = \frac{\text{# of individuals having a certain attribute in the sample}}{\text{sample size}} = \frac{\text{# of successes in the sample}}{n}.[/latex]
Here are several examples:
Examples
- A random sample of n = 100 students is obtained from MacEwan University. Of the 100 students in the sample, 65 are female. The sample proportion [latex]\hat{p} = \frac{x}{n} = \frac{65}{100}[/latex] provides a point estimate of [latex]p[/latex], the proportion of female students at MacEwan.
- A random sample of n = 1000 judo matches is obtained, and it is determined that 510 of the matches are won by the athletes wearing a blue uniform. The sample proportion [latex]\hat{p} = \frac{x}{n} = \frac{510}{1000}[/latex] is a point estimate of [latex]p[/latex], the proportion of winners in blue.
- A credit card company sends an advertisement to n = 500 randomly chosen customers and only 10 customers respond. The sample proportion [latex]\hat{p} = \frac{x}{n} = \frac{10}{500}[/latex] is a point estimate of [latex]p[/latex], the proportion of respondents.
