Sampling Variability: Learn It 4

  • Check the conditions for normal approximation of a sampling distribution of a sample proportion.
  • Use the normal distribution to calculate probabilities and percentiles from a sampling distribution.
  • Find the sample size needed for a sampling distribution to have a desired standard deviation.

Calculate the Sample Size [latex]n[/latex]

Recall that the standard deviation of the sampling distribution for a sample proportion can be calculated as [latex]\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}[/latex]. If the researchers want to limit their standard deviation, or variability, then they can use this formula to find the sample size instead.

[latex]\begin{array}{l l} \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} &\quad& \text{Start with the standard deviation formula. } \\ (\sigma_{\hat{p}})^2 = \frac{p(1-p)}{n} && \text{Square both sides to eliminate the square root.} &&\\ n \cdot (\sigma_{\hat{p}})^2 = p(1-p) && \text{Multiply both sides by } n \text{ to isolate } p(1-p) \text{ on one side.} \\ n = \frac{p(1-p)}{(\sigma_{\hat{p}})^2} && \text{Divide both sides by } (\sigma_{\hat{p}})^2 \text{ to solve for } n. \end{array}[/latex]

Sample Size and Normal Distribution for a Sample Proportion