- Use technology to create a sampling distribution of a sample proportion given [latex]n[/latex] and [latex]p[/latex].
- Calculate the mean and standard deviation for a sampling distribution of a sample proportion.
- Recognize the difference between the standard deviation and the standard error of a sample proportion.
Standard Deviation vs. Standard Error
The previous exercise assumed that we knew the value of the population proportion and could either calculate the mean and standard deviation of the sample proportion by formulas, or estimate the mean and standard deviation by simulation. In practice, we do not typically know the population proportion, nor do we have the luxury of taking thousands of random samples. Instead, we observe a single random sample. In this case, we need to estimate the mean and standard deviation of the sample proportion.
The estimated mean and standard deviation of the sampling distribution of a sample proportion
- The estimated mean of the distribution of sample proportions is [latex]\hat{p}[/latex].
- To distinguish it from the true standard deviation of sample proportions, we call the estimated standard deviation of sample proportions the standard error of [latex]\hat{p}[/latex]:
Standard Error: [latex]SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}[/latex]
Simulation provides one way to estimate the standard deviation of the sample proportion, and this formula gives another way.