Often in statistics we’re interested in estimating the proportion of individuals in a population with a certain characteristic.
For example, we might be interested in estimating the proportion of residents in a certain city who support a new law.
Instead of going around and asking every individual resident if they support the law, we would instead collect a simple random sample and find out how many residents in the sample support the law.
We would then calculate the sample proportion (p̂) as:
Sample Proportion Formula:
p̂ = x / n
- x: The count of individuals in the sample with a certain characteristic.
- n: The total number of individuals in the sample.
We would then use this sample proportion to estimate the population proportion. For example, if 47 of the 300 residents in the sample supported the new law, the sample proportion would be calculated as 47 / 300 = 0.157.
This means our best estimate for the proportion of residents in the population who supported the law would be 0.157.
However, there’s no guarantee that this estimate will exactly match the true population proportion so we typically calculate the standard error of the proportion as well.
This is calculated as:
Standard Error of the Proportion Formula:
Standard Error = √p̂(1-p̂) / n
For example, if p̂ = 0.157 and n = 300, then we would calculate the standard error of the proportion as:
Standard error of the proportion = √.157(1-.157) / 300 = 0.021
We then typically use this standard error to calculate a confidence interval for the true proportion of residents who support the law.
This is calculated as:
Confidence Interval for a Population Proportion Formula:
Confidence Interval = p̂ +/- z*√p̂(1-p̂) / n
Looking at this formula, it’s easy to see that the larger the standard error of the proportion, the wider the confidence interval.
Note that the z in the formula is the z-value that corresponds to popular confidence level choices:
For example, here’s how to calculate a 95% confidence interval for the true proportion of residents in the city who support the new law:
- 95% C.I. = p̂ +/- z*√p̂(1-p̂) / n
- 95% C.I. = .157 +/- 1.96*√.157(1-.157) / 300
- 95% C.I. = .157 +/- 1.96*(.021)
- 95% C.I. = [ .10884 , .19816]
Thus, we would say with 95% confidence that the true proportion of residents in the city who support the new law is between 10.884% and 19.816%.