Statistics / Unit 8: Sampling Distributions and Confidence Intervals
8.2 Sampling Distribution of a Sample Proportion
Notes
For a sample proportion $\hat{p}$ from a population with proportion $p$:
- Mean: $\mu_{\hat{p}} = p$ (the statistic is unbiased).
- Standard deviation: $\displaystyle \sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$.
- The shape is approximately normal when $np \ge 10$ and $n(1 - p) \ge 10$.
- Quadrupling $n$ cuts the standard deviation in half.
- Example: $p = 0.5$ and $n = 100$ give $\displaystyle \sigma_{\hat{p}} = \sqrt{\frac{0.25}{100}} = 0.05$.
Practice
For $p = 0.5$ and $n = 100$, the standard deviation of $\hat{p}$ is:
$\displaystyle \sqrt{\frac{0.5 \cdot 0.5}{100}} = \sqrt{0.0025} = 0.05$.
The sample size changes from 100 to 400. The standard deviation of $\hat{p}$ is multiplied by about:
$\sigma_{\hat{p}}$ has $\displaystyle \sqrt{n}$ in the denominator, and $\displaystyle \sqrt{4} = 2$.
The true proportion of a population is $p = 0.35$. The mean of the sampling distribution of $\hat{p}$ is:
$\mu_{\hat{p}} = p$, regardless of $n$.
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