AP Stats 3.2: Sampling Distribution of p-hat

What topic 3.2 covers

Topic 3.2 describes how the sample proportion $\hat{p}$ behaves when you take many samples of the same size $n$ from a population with true proportion $p$. That pattern of possible $\hat{p}$ values is the sampling distribution of the sample proportion. You learn its center, its spread, and the conditions that make it roughly normal, which is what every later proportion procedure relies on. Keep three distributions separate in your mind: the population, one sample of data, and the sampling distribution of a statistic from repeated samples.

Mean and standard deviation

When the sampled values are independent, the sampling distribution of $\hat{p}$ has mean $\mu_{\hat{p}} = p$ and standard deviation $\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}$. The mean equals $p$, which restates that $\hat{p}$ is an unbiased estimator from topic 3.1. The standard deviation shrinks as $n$ grows, so larger samples cluster more tightly around $p$. Because $n$ sits under the square root, quartering the spread takes about sixteen times the sample size, not four times.

The three conditions

Two conditions justify using the standard deviation formula, and one more justifies the normal shape.

1. Randomization: the data come from a random sample.
2. 10% condition: when sampling without replacement, the population is at least 10 times the sample size, so $n \le 0.10N$.
3. Normality (large counts): the expected successes $np$ and expected failures $n(1-p)$ are both at least 10.

The 10% condition keeps the sampled values close enough to independent, and the large-counts condition keeps the distribution symmetric enough to model with a normal curve. When all three hold, you can compute probabilities for $\hat{p}$ using the normal distribution and a z-score.

Interpreting the values

Always read the mean, standard deviation, and any probability in the context of a specific population. The standard deviation is the typical distance a single sample proportion falls from $p$. A probability from this distribution answers a question like "how often would a sample proportion this far from $p$ occur by chance," which is exactly the reasoning behind the p-values in topic 3.6.

This topic is also the engine behind the confidence intervals of topic 3.3, since the interval width comes directly from this spread. Everything downstream in Unit 3, from margins of error to test statistics, traces back to the mean and standard deviation described here.