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STA260 Lecture 02

STA260 Lecture 02 Raw
STA260 Post-Lecture 02

• Overview

  • This lecture covers the sampling distributions of the mean, specifically when the population is Normal versus when it is not (requiring CLT).
  • It introduces the derivation of the Chi-squared distribution from the Standard Normal.

• Topics

  • Sampling Distribution of Sample Mean (Normal Population)
    • If the population is Normal, the sample mean is exactly Normal.
  • Proof - MGF of Sample Mean from Normal Population
    • Derivation using the properties of Moment Generating Functions.
  • Central Limit Theorem
    • Application to non-normal populations (e.g., Continuous Data).
  • Relationship between Standard Normal and Chi-squared
    • How squaring a standard normal variable leads to a Chi-squared distribution.

• Examples

  • Beer Bottling (Normal Population)
    • See: Sampling Distribution of Sample Mean (Normal Population)
  • Navy Exchange (CLT Application)
    • Context: Service times are iid with $\mu =1.5$ and $\sigma =1$.
    • Objective: Find probability that $n=100$ customers are serviced in $<120$ minutes.
    • Calculation:
      • $P(\sum X_i\le 120)=P(\overline{X}\le 1.2)$.
      • By Central Limit Theorem, $\overline{X}\approx N(1.5,0.01)$.
      • Standardize: $Z=\frac{1.2-1.5}{0.1}=-3$.
      • Using Empirical Rule, $P(Z\le -3)\approx 0.0015$.

• Footnotes