Distribution of Sample Proportion $\hat{p}$

For a sample proportion $\hat{p}$:

Mean of the sampling distribution: $\mu_{\hat{p}} = p$

The sample proportion centers on the true population proportion.

Standard Error (SE) of $\hat{p}$: $SE(\hat{p}) = \sqrt{\frac{p(1-p)}{n}}$

Conditions for approximation by normal distribution:

• $np \geq 10$ (at least 10 successes)
• $n(1-p) \geq 10$ (at least 10 failures)
• Sample is random
• 10% rule: n ≤ 0.10N (sample ≤ 10% of population)

Distribution of Sample Mean $\bar{x}$

For a sample mean $\bar{x}$:

Mean of the sampling distribution: $\mu_{\bar{x}} = \mu$

The sample mean centers on the true population mean.

Standard Error (SE) of $\bar{x}$: $SE(\bar{x}) = \frac{\sigma}{\sqrt{n}}$

where σ is the population standard deviation.

Conditions for approximation by normal distribution:

• If population is normal: any sample size works
• If population shape unknown: n ≥ 30 (Central Limit Theorem)
• Sample is random
• 10% rule: n ≤ 0.10N

Key Properties of Sampling Distributions

1. Center: Both $\hat{p}$ and $\bar{x}$ are unbiased (centered on true parameter)
2. Spread: SE decreases as n increases; larger samples give less variable statistics
3. Shape: Approximately normal under appropriate conditions
4. Variability formula: SE depends on population variability and sample size

Worked Example

Suppose 40% of customers prefer Brand A. You take a random sample of 100 customers.

For $\hat{p}$:

• $\mu_{\hat{p}} = 0.40$
• $SE(\hat{p}) = \sqrt{\frac{0.40 \cdot 0.60}{100}} = \sqrt{\frac{0.24}{100}} = \sqrt{0.0024} = 0.049$
• Check conditions: np = 40 ≥ 10 ✓, n(1−p) = 60 ≥ 10 ✓

The sampling distribution of $\hat{p}$ is approximately $N(0.40, 0.049^2)$.

Common Mistakes

1. Confusing SE with standard deviation: SE is smaller than σ because of the $\sqrt{n}$ in denominator
2. Forgetting conditions: Always verify the sample size conditions before using normal approximation
3. Not recognizing center: Sample statistics are unbiased; they center on the true parameter

AP Exam Tip

Sampling distribution questions require you to identify whether you're working with $\bar{x}$ or $\hat{p}$, then apply correct formula. Know the SE formulas and always check conditions. If conditions fail, state the issue rather than proceeding with the normal approximation.