Sampling Distribution of p-hat

The distribution of p-hat

• A sample proportion 样本比例 $\hat{p}$ estimates the true population proportion $p$.
• Its sampling distribution is centered at $p$: $\mu_{\hat{p}} = p$ (so $\hat{p}$ is unbiased).
• Different samples give different $\hat{p}$ — the spread is what we quantify next.
• Under the right conditions, this distribution is approximately normal.

Its standard deviation

• The standard deviation of $\hat{p}$ is:
• $$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$
• Bigger $n$ → smaller spread → a more precise estimate (the $\sqrt{n}$ in the bottom).
• This measures the typical distance of $\hat{p}$ from the true $p$.

The sampling distribution of p̂ is centred at p with standard deviation √(p(1−p)/n)

The large counts condition

• To use a normal model for $\hat{p}$, check the large counts condition 大计数条件:
• $$np \ge 10 \quad\text{and}\quad n(1-p) \ge 10$$
• In words: expect at least $10$ successes and at least $10$ failures.
• If either count is under $10$, the sampling distribution is too skewed for the normal model.

The 10% condition

• When sampling without replacement, check the 10% condition 10%条件.
• The sample must be no more than $10\%$ of the population: $n \le 0.10 N$.
• This keeps the trials "close enough" to independent for the SD formula to hold.
• A small slice of a big population barely changes as you draw.

Write from a standard deviation of 0.04

• The population proportion is 0.2.
• The sample size is 100. The lesson standard deviation 0.05 is another sample size or proportion.