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

1. Probability about x̄: Find P(x̄ < a), P(x̄ > b), given μ, σ, n
2. Finding sample mean values: Given probability, find cutoff
3. CLT verification: Checking conditions, applying theorem
4. 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

Related Course Page

• FAD1015 - Mathematics III