1. The problem involves understanding the formula for the variance of the sampling distribution of the sample proportion $\hat{p}$. 2. The formula given is $$\text{Var}(\hat{p}) = \frac{\hat{p}(1-\hat{p})}{n}$$ where $\hat{p}$ is the sample proportion and $n$ is the sample size. 3. This formula tells us how the variability of the sample proportion depends on the proportion itself and the size of the sample. 4. Important rules: - $\hat{p}$ must be between 0 and 1 because it is a proportion. - $n$ must be a positive integer representing the number of observations. 5. The variance decreases as $n$ increases, meaning larger samples give more precise estimates of the population proportion. 6. The term $\hat{p}(1-\hat{p})$ is maximized when $\hat{p} = 0.5$, so the variance is largest when the proportion is around 0.5 and smaller near 0 or 1. 7. This formula is fundamental in statistics for constructing confidence intervals and hypothesis tests about proportions. Final answer: The variance of the sample proportion $\hat{p}$ is $$\boxed{\frac{\hat{p}(1-\hat{p})}{n}}$$ which quantifies the spread of the sampling distribution of $\hat{p}$.