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STA258 Lecture 10
STA258 Lecture 10 Raw
STA258 Lecture 10 Flashcards
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Completed Notes Status
- Completed insertions: 9
- Ambiguities left unresolved: 4
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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
when assuming and independent normal samples. - Degrees of freedom:
; interpret sign of CI for direction of mean difference.
- Use pooled variance
- Welch's t-test:
- Use unpooled standard error when
(or equality is doubtful). - Use Welch–Satterthwaite degrees of freedom; conservative alternative is
.
- Use unpooled standard error when
- Two-Proportion Z Confidence Interval:
- Estimate
with and use a critical value when large-sample conditions hold. - Always align the algebraic sign with the verbal claim (which group is "1" vs "2").
- Estimate
- Two-Sample t Confidence Interval (Equal Variances):
- 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.
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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
CI does not mean there is a chance that ; the CI is a procedure that would cover the fixed parameter in of repeated samples. - To quantify evidence for "direct is better," use a one-sided framing: test
vs (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.
- A
- Answer:
- #tk: "How sure can we be that direct is better than broker?" (and "95/100 times we will have a better return directly.")
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Practice Questions
- Remember/Understand:
- What assumptions justify using the pooled-variance two-sample t CI (and what is the df)?
- In a CI for
, what does it mean if 0 is inside the interval vs outside? - Why do difference-of-proportions CIs typically use
instead of ?
- Apply/Analyze:
- Compute a
CI for using pooled variance, given . - For a two-proportion dataset, build a CI for
and write a one-sentence interpretation that correctly matches the sign. - For paired data, compute differences
and build a CI for ; explain why pairing changes the analysis.
- Compute a
- 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.
- Remember/Understand:
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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,
placement). - Study strategy: Memorize the structure (variance-over-
terms), then verify with R output on a small synthetic example.
- Why it's challenging: The Welch df formula is algebra-heavy and easy to misapply (variance vs standard deviation,
- Confidence Intervals:
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Action Plan
- Immediate review actions:
- Practice and application:
- Deep dive study:
- Verification and integration:
- Immediate review actions:
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Footnotes