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STA260 Lecture 06

STA260 Lecture 06 Raw
STA260 Lecture 06 Flashcards

• Review of Standard Normal and Chi-Squared

  • Standardizing a sample mean:
    • $Z=\frac{\overline{X}-\mu }{\frac{\sigma }{\sqrt{n}}}\sim N(0,1)$.
  • Sum of squares connection:
    • If $Z_i\stackrel{iid}{\sim }N(0,1)$, then $\sum Z_i^2\sim {\chi }_{(n)}^2$.^[1]
    • See: Chi-squared from Sum of Squared Standard Normals.
    • See: Sum of Squares is Chi-squared.

• T Distribution

  • Definition:
    • Ratio of a standard normal to the square root of a scaled Chi-squared.
    • See: T Distribution.
  • Derivation from sampling statistics:
    • We transform the standardized sample mean (unknown $\sigma$) into the $T$ definition.
    • $\frac{\overline{X}-\mu }{S/\sqrt{n}}\sim t_{(n-1)}$.
    • See: Derivation of t-statistic for Sample Mean.
  • Key Components:
    • Numerator: Standard Normal $Z$.
    • Denominator: $\sqrt{{\chi }_{(n-1)}^2/(n-1)}$.
    • Independence: $\overline{X}\perp S^2$ (essential for the ratio).^[2]

• F Distribution

  • Definition:
    • Ratio of two independent Chi-squared variables, each divided by their degrees of freedom.
    • $F=\frac{U/d_1}{V/d_2}$.^[3]
    • See: F Distribution.
  • Application to Variance Ratios:
    • Used to compare population variances ${\sigma }_1^2$ and ${\sigma }_2^2$.
    • Statistic: $\frac{S_1^2}{S_2^2}$ (under $H_0:{\sigma }_1={\sigma }_2$).
    • Relies on: Marginal Distribution of Sample Variance.

• Lecture Summary

  • Workflow: Standardize to known distributions.
    • Mean (known $\sigma$) $\to$ Normal ($Z$).
    • Variance $\to$ Chi-squared (${\chi }^2$).
    • Mean (unknown $\sigma$) $\to$ Student's $t$ ($t$).
  • Synthesis:
    • The $t$-distribution arises naturally when we replace $\sigma$ with $S$ in the Z-score.
    • This substitution introduces extra variability (in the denominator), "fattening" the tails.^[2:1]
  • Variance Ratios:
    • Ratios of sample variances lead naturally to the $F$ distribution because sample variances are scaled ${\chi }^2$ variables.^[3:1]

• Practice Questions

  • Remember/Understand:
    • Define Student's $t$ using $Z$ and $V$.
    • Distribution of $\sum Z_i^2$.^[1:1]
    • Explain why $S^2$ uses $n-1$ degrees of freedom.
  • Apply/Analyze:
    • Rewrite $\frac{\overline{X}-\mu }{S/\sqrt{n}}$ into the form $Z/\sqrt{U/\nu }$.
    • Compare/Contrast: When to use $Z$ vs $t$ for inference on $\mu$.
  • Evaluate/Create:
    • Checklist for choosing $Z$, ${\chi }^2$, $t$, or $F$.
    • Identify assumptions: Where does independence enter the $t$-derivation?

• Challenging Concepts

  • Degrees of Freedom in Sample Variance:
    • Why $n-1$?
    • The linear constraint $\sum (X_i-\overline{X})=0$ reduces the dimension of the data vector space by 1.
  • Structure of the t-statistic:
    • It is purely algebraic manipulation to get from the sample statistic to the theoretical definition $Z/\sqrt{U/\nu }$.
  • Independence Assumptions:
    • $t$ and $F$ derivations fail if the numerator and denominator are not independent.
    • For normal samples, $\overline{X}\perp S^2$ ensures this holds.

• Action Plan

  • Immediate:
    • Review lines marked (COMPLETED) in raw notes.
    • Scan for missing #tk items.
  • Deep Dive:
    • Drill the geometry of $n-1$ (subspace projection).
    • Practice the algebraic rewrite of $t$ until fluent.
  • Verification:
    • Cross-reference independence claims with textbook.

• Footnotes