1. Sampling Distribution

Definition: A sampling distribution is the probability distribution of a statistic (like mean or proportion) obtained from all possible samples of a fixed size (n) drawn from a population.

• It shows how a sample statistic varies from sample to sample.
• Central in inferential statistics for estimating population parameters.

Key Concept:

• Population → Take samples → Compute statistic (mean, proportion) → Distribution of statistic = Sampling Distribution

2. Sampling Distribution of the Mean

Scenario:

• Population has mean $\mu$ and standard deviation $\sigma$
• Sample size = $n$

Central Limit Theorem (CLT):

• For a sufficiently large sample ($n \ge 30$), the sampling distribution of the sample mean $\bar{X}$ is approximately normal, regardless of population distribution.

Properties of Sampling Distribution of $\bar{X}$:

Example:

• Population mean height = 165 cm, $\sigma = 10$ cm, sample size $n = 25$ $$\sigma_{\bar{X}} = \frac{10}{\sqrt{25}} = 2 \text{ cm}$$
• The sample mean $\bar{X}$ will vary around 165 cm with SD = 2 cm.

3. Sampling Distribution of Proportion

Scenario:

• Population proportion of success = $p$
• Sample size = $n$

Sample Proportion:

Properties:

• Mean: $E(\hat{p}) = p$
• Standard Error: $$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$$
• For large $n$, $\hat{p}$ is approximately normally distributed (Normal approximation of Binomial)

Example:

• Population proportion of smokers = 0.3, sample size $n = 100$ $$\sigma_{\hat{p}} = \sqrt{\frac{0.3(1-0.3)}{100}} = \sqrt{0.0021} \approx 0.0458$$

4. Key Points to Remember

1. Sampling distribution shows the variability of a statistic across samples.

2. Mean of sampling distribution = population mean ($E(\bar{X}) = \mu$, $E(\hat{p}) = p$)

3. Standard Error (SE) measures variability:

   • Mean: $\sigma_{\bar{X}} = \sigma/\sqrt{n}$
   • Proportion: $\sigma_{\hat{p}} = \sqrt{p(1-p)/n}$

4. Shape:

   • Large $n$ → approximately normal (Central Limit Theorem)

5. Larger sample size → smaller standard error → more precise estimate

5. Summary Table