FAD1015 Tutorial 9 — Sampling Methods
Tutorial questions on sampling distributions, Central Limit Theorem applications, and sampling methodology. Source file: FAD1015 25-26 Tutorial 9 Questions.pdf
Summary
Problem set focused on sampling distributions of the mean, Central Limit Theorem applications, and calculations involving standard error.
Topics Covered
1. Sampling Distribution Concepts
- Sample mean as random variable
- Distribution of x̄
- Standard error: σ/√n
2. Central Limit Theorem Applications
- When CLT applies (n ≥ 30 rule of thumb)
- Finding probabilities about sample means
- Non-normal populations → normal sampling distribution
3. Sampling Distribution Properties
- E[x̄] = μ
- Var(x̄) = σ²/n
- Shape approaches normal as n increases
4. Finite Population Correction
- When n/N > 0.05
- Correction factor: √((N-n)/(N-1))
Key Formulas
Sampling Distribution of Mean: $$\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \text{ for large } n$$
Standard Error: $$SE = \frac{\sigma}{\sqrt{n}}$$
Z-Score for Sample Mean: $$Z = \frac{\bar{X} - \mu}{\sigma/\sqrt{n}}$$
Finite Population Correction: $$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \cdot \sqrt{\frac{N-n}{N-1}}$$
Problem Types
- Probability about x̄: Find P(x̄ < a), P(x̄ > b), given μ, σ, n
- Finding sample mean values: Given probability, find cutoff
- CLT verification: Checking conditions, applying theorem
- Finite population: Applying correction factor
Related Lectures
- FAD1015 L20 — Sampling Distribution of the Mean
- FAD1015 L21-L22 — Estimation of Population Mean — builds on sampling concepts
Related Concepts
- Probability Distributions
- Hypothesis Testing — sampling distributions form foundation