FAD1015 Tutorial 10 — Estimation of Population Mean

Tutorial questions on confidence intervals for population mean with known and unknown population standard deviation. Source file: FAD1015 25-26 Tutorial 10 Questions.pdf

Summary

Problem set covering point estimation, confidence interval construction, interpretation, and sample size determination for estimating population mean.

Topics Covered

1. Point Estimation

• Sample mean as point estimator
• Properties: unbiasedness, consistency

2. Confidence Intervals (σ Known)

• Formula: x̄ ± z_(α/2) × σ/√n
• Critical values from standard normal
• Margin of error calculation

3. Confidence Intervals (σ Unknown)

• Using t-distribution
• Degrees of freedom: n - 1
• Formula: x̄ ± t_(α/2,n-1) × s/√n
• t-table usage

4. Sample Size Determination

• n = (z_(α/2) × σ / E)²
• Balancing confidence level and precision

5. Interpretation

• Correct vs incorrect interpretations
• Confidence level meaning

Key Formulas

σ Known: $$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$

σ Unknown: $$\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$$

Sample Size: $$n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2$$

Margin of Error: $$E = \text{critical value} \times \text{standard error}$$

Problem Types

1. CI with σ known: Given x̄, σ, n, construct interval
2. CI with σ unknown: Given sample data, use t-distribution
3. Sample size: Determine n for desired margin of error
4. Interpretation: Explain meaning of confidence level

Related Lectures

• FAD1015 L21-L22 — Estimation of Population Mean
• FAD1015 L20 — Sampling Distribution of the Mean — prerequisite
• FAD1015 L19 — Input Data & Descriptive Statistics in R — R implementation

Related Concepts

• Probability Distributions
• Hypothesis Testing — uses similar test statistic concepts

Related Course Page

• FAD1015 - Mathematics III