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STA258 Lecture 10

STA258 Lecture 10 Raw
STA258 Lecture 10 Flashcards

• Completed Notes Status

  • Completed insertions: 9
  • Ambiguities left unresolved: 4

• Lecture Summary

  • Central objective: Build and interpret Confidence Intervals for differences (two means, paired means, two proportions, and two variances) while tracking assumptions and practical meaning.
  • Key concepts:
    • Two-Sample t Confidence Interval (Equal Variances):
      • Use pooled variance $S_p^2$ when assuming ${\sigma }_1^2={\sigma }_2^2$ and independent normal samples.
      • Degrees of freedom: $\nu =n_1+n_2-2$; interpret sign of CI for direction of mean difference.
    • Welch's t-test:
      • Use unpooled standard error when ${\sigma }_1^2\ne {\sigma }_2^2$ (or equality is doubtful).
      • Use Welch–Satterthwaite degrees of freedom; conservative alternative is $\nu =min(n_1-1,n_2-1)$.
    • Two-Proportion Z Confidence Interval:
      • Estimate $p_1-p_2$ with ${\hat{p}}_1-{\hat{p}}_2$ and use a $z^{\star }$ critical value when large-sample conditions hold.
      • Always align the algebraic sign with the verbal claim (which group is "1" vs "2").
  • Connections:
    • "CI excludes 0" is a statistical signal of non-zero difference, while distance from 0 (and units) drives Practical vs Statistical Significance.
    • Assumption checks (e.g., Normal Q-Q Plot) protect inference validity for small-sample mean procedures.

• TK Resolutions

  • #tk: "How sure can we be that direct is better than broker?" (and "95/100 times we will have a better return directly.")
    • Answer:
      • A $95\mathrm{\%}$ CI does not mean there is a $95\mathrm{\%}$ chance that ${\mu }_d>{\mu }_b$; the CI is a procedure that would cover the fixed parameter ${\mu }_d-{\mu }_b$ in $95\mathrm{\%}$ of repeated samples.
      • To quantify evidence for "direct is better," use a one-sided framing: test $H_0:{\mu }_d-{\mu }_b\le 0$ vs $H_A:{\mu }_d-{\mu }_b>0$ (or report a one-sided CI / p-value), and also assess whether the lower CI bound is meaningfully above 0 in domain units.
      • The phrase "95/100 investors" is about individual outcomes (a probability for a random investor beating another), which is not what a CI for mean difference answers; that would need a distributional model for individual returns, not only mean parameters.

• Practice Questions

  • Remember/Understand:
    • What assumptions justify using the pooled-variance two-sample t CI (and what is the df)?
    • In a CI for ${\mu }_1-{\mu }_2$, what does it mean if 0 is inside the interval vs outside?
    • Why do difference-of-proportions CIs typically use $z^{\star }$ instead of $t^{\star }$?
  • Apply/Analyze:
    • Compute a $95\mathrm{\%}$ CI for ${\mu }_1-{\mu }_2$ using pooled variance, given $n_1,n_2,{\overline{x}}_1,{\overline{x}}_2,s_1^2,s_2^2$.
    • For a two-proportion dataset, build a CI for $p_1-p_2$ and write a one-sentence interpretation that correctly matches the sign.
    • For paired data, compute differences $d_i$ and build a CI for ${\mu }_d$; explain why pairing changes the analysis.
  • Evaluate/Create:
    • Given two groups with unequal variances and unequal sample sizes, decide between pooled t, Welch, or a conservative df approach, and justify your choice using assumptions and consequences.

• Challenging Concepts

  • Confidence Intervals:
    • Why it's challenging: A CI is often misread as a probability statement about the parameter, or as a statement about individual outcomes.
    • Study strategy: Write the "repeated sampling" interpretation verbatim, then practise translating CI endpoints into domain-specific meaning and units.
  • Welch's t-test:
    • Why it's challenging: The Welch df formula is algebra-heavy and easy to misapply (variance vs standard deviation, $n$ placement).
    • Study strategy: Memorize the structure (variance-over-$n$ terms), then verify with R output on a small synthetic example.

• Action Plan

  • Immediate review actions:
    • Review all [COMPLETED] insertions and verify with course materials
    • Address all #tk items (if any)
    • Re-read Lecture Summary and confirm key links
  • Practice and application:
    • Answer Remember/Understand questions
    • Answer Apply/Analyze questions
    • Attempt Evaluate/Create questions
  • Deep dive study:
    • Focus on Challenging Concepts (above)
    • Extend atomic notes only where the new lecture adds content
  • Verification and integration:
    • Cross-reference notes with textbook/slides
    • Identify remaining gaps
    • Link this lecture to prior/future lecture notes (lecture notes only; no MOCs)

• Footnotes