- Check the conditions for creating a confidence interval for population proportion.
- Describe the connection between the confidence level and the confidence interval.
- Calculate a confidence interval for a population proportion.
When taking random samples of size [latex]n[/latex] from a population distribution with proportion [latex]p[/latex]:
- The mean of the distribution of sample proportions is [latex]p[/latex].
- The standard deviation of the distribution of sample proportions is[latex]\sqrt{\frac{p(1-p)}{n}}[/latex].
- If [latex]np \geq 10[/latex] and [latex]n(1-p) \geq 10[/latex], then the Central Limit Theorem (CLT) states that the distribution of the sample proportions follows an approximate normal distribution with mean [latex]p[/latex] and standard deviation [latex]\sqrt{\frac{p(1-p)}{n}}[/latex].
Standard error
When the sample size is large enough, we can use [latex]\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/latex] in place of[latex]\sqrt{\frac{p(1-p)}{n}}[/latex]. This is called the standard error, the estimated standard deviation of sample proportions. It is the measure of sample-to-sample variability. We will use the standard error to help us convey information about the accuracy of our point estimate.