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STA260 Lecture 03

STA260 Lecture 03 Raw
STA260 Post-Lecture 03

• Overview

  • This lecture covers the exact sampling distribution for normal populations, the standardization process for sample means, and methods for determining necessary sample sizes.
  • It also reviews convergence concepts and the Central Limit Theorem from STA256.

• Exact Sampling Distributions

  • Core Concept: Sampling Distribution of Gaussian Means
  • If the population is normal, the sample mean is exactly normal regardless of sample size.
  • This contrasts with the Central Limit Theorem, which is an approximation for large $n$.

• Standardization and Probabilities

  • Core Concept: Standardizing Sample Means
  • Process for converting $\overline{X}$ to $Z\sim N(0,1)$ to calculate probabilities.
  • Key difference from standard variables: Scaling by standard error $\frac{\sigma }{\sqrt{n}}$ rather than $\sigma$.

• Sample Size Determination

  • Core Concept: Sample Size for Mean Precision
  • Method for calculating the minimum $n$ required to estimate $\mu$ within a specific margin of error and confidence level.

• Review: Convergence and CLT

  • Concept: Convergence in Distribution
    • Definition: $X_n\stackrel{D}{\to }X$ if $\underset{n\to \mathrm{\infty }}{lim}{F}_{X_n}(x)=F_X(x)$.
  • Theorem: Central Limit Theorem
    • Application: The distribution of means approaches normality for any population with finite variance.
    • Result: $\overline{X}\stackrel{D}{\to }N(\mu ,\frac{{\sigma }^2}{n})$.
  • Example: Service Times
    • Problem: Approximate probability that $n=100$ customers can be served in 2 hours ($120$ mins).
    • Given: $\mu =1.5$ mins, $\sigma =1$ min.
    • Setup: $P(\sum _{i=1}^{100}{X}_i<120)$.
    • Conversion to Mean: $P(n\overline{X}<120)⟹P(\overline{X}<1.2)$.
    • Standardization:
      • $$P(\frac{\overline{X}-1.5}{1/\sqrt{100}}<\frac{1.2-1.5}{1/\sqrt{100}})$$
      • $$P(10(\overline{X}-1.5)<-3)$$
      • $$P(Z<-3)=0.00135\$\$(bysymmetry).$$