STA258 Lecture 09

Completed Notes Status

Central objective: Extend confidence interval methodology to compare two independent population means when variances are equal, using the Pooled Variance Estimator and T-Distribution.
Key concepts:
- Two-Sample T-Interval for Means: Constructs confidence intervals for the difference in means ( $μ_{1} - μ_{2}$ ) when population variances are equal but unknown
- Pooled Variance Estimator: A weighted average of two sample variances that provides a single estimate of the common population variance, denoted $S_{p}^{2}$ ^[1]
- Convex Combination: The pooled estimator is a convex combination where weights sum to 1, ensuring the estimate lies between the two sample variances
- T-Distribution: Used when replacing unknown $σ$ with $S_{p}$ , accounting for additional uncertainty with degrees of freedom $n_{1} + n_{2} - 2$
Connections:
- The Central Limit Theorem justifies that $\hat{Δ} = \bar{Y} - \bar{X}$ follows a normal distribution for large samples
- Unbiased estimation: The pooled variance estimator is unbiased for $σ^{2}$ when $σ_{1}^{2} = σ_{2}^{2}$ ^[1:1]
- Independence of Sample Mean and Sample Variance within each sample allows construction of the T-statistic
- Hypothesis testing: When 0 is not in the CI for $μ_{1} - μ_{2}$ , we reject equality of means at the given confidence level

#tk: Bonus question answer
- Answer: [COMPLETED] Song is Purple Rain by Prince
#tk: Is this note accurate about plug-in principle?
- Answer: Yes, when $σ$ is unknown, we replace it with an estimator (sample standard deviation or pooled standard deviation) following the plug-in principle. This substitution changes the sampling distribution from normal to T-distribution.
#tk: He skipped too many steps in unbiasedness proof
- Answer: The key steps omitted were: After applying linearity of expectation and substituting $E [S_{1}^{2}] = σ^{2}$ and $E [S_{2}^{2}] = σ^{2}$ , we get $\frac{n_{1} - 1}{n_{1} + n_{2} - 2} σ^{2} + \frac{n_{2} - 1}{n_{1} + n_{2} - 2} σ^{2} = σ^{2} (\frac{n_{1} - 1 + n_{2} - 1}{n_{1} + n_{2} - 2}) = σ^{2} (\frac{n_{1} + n_{2} - 2}{n_{1} + n_{2} - 2}) = σ^{2}$ . This completes the proof that $E [S_{p}^{2}] = σ^{2}$ ^[1:2].
#tk: Is this a case where we can jack up sample numbers to achieve a better CI interval?
- Answer: Increasing sample size will narrow the confidence interval, providing more precision. However, in this case, the interval already excludes 0, so larger samples would only strengthen the conclusion we already have. The main benefit of larger $n$ is reducing the width of the interval for future studies^[2].

Remember/Understand:
- Why must we assume equal variances ( $σ_{1}^{2} = σ_{2}^{2}$ ) to use the pooled variance estimator?
- What are the degrees of freedom for the T-distribution when constructing a two-sample pooled confidence interval?
- Define a convex combination and explain why the pooled variance is one.
Apply/Analyze:
- Given two samples with $n_{1} = 20, S_{1}^{2} = 50$ and $n_{2} = 30, S_{2}^{2} = 70$ , compute $S_{p}^{2}$ .
- If a 95% CI for $μ_{1} - μ_{2}$ is $[- 10, - 2]$ , what can you conclude about the relationship between the two population means?
- Verify that the weights $\frac{n_{1} - 1}{n_{1} + n_{2} - 2}$ and $\frac{n_{2} - 1}{n_{1} + n_{2} - 2}$ sum to 1 for arbitrary $n_{1}, n_{2} > 1$ .
Evaluate/Create:
- A researcher computes a pooled T-interval but later discovers the population variances differ substantially. Evaluate the validity of the confidence interval and propose an alternative approach.
- Design a study to compare mean verbal scores between two student groups. Specify sample size, assumptions, and how you would verify equal variance assumption before constructing the interval.

Pooled Variance Estimator:
- Why it's challenging: Understanding why we weight by $(n - 1)$ rather than $n$ , and recognizing that pooling is only valid when $σ_{1}^{2} = σ_{2}^{2}$ .
- Study strategy: Work through the unbiasedness proof step-by-step, focusing on how the weights form a convex combination and why this matters. Practice computing $S_{p}^{2}$ manually to build intuition for how sample sizes affect weighting.
T-Distribution with pooled variance:
- Why it's challenging: Keeping track of which quantities are independent (sample means vs pooled variance) and correctly identifying degrees of freedom as $n_{1} + n_{2} - 2$ instead of $n_{1} + n_{2} - 1$ .
- Study strategy: Draw a diagram showing the independence structure: $\bar{Y} ⊥ S_{1}^{2}$ , $\bar{X} ⊥ S_{2}^{2}$ , and therefore $(\bar{Y} - \bar{X}) ⊥ S_{p}^{2}$ . Memorize the degrees of freedom formula and understand it as "total observations minus parameters estimated."

Immediate review actions:
- Review all [COMPLETED] insertions and verify with course materials
- Address all #tk items (all resolved above)
- Re-read Lecture Summary and confirm key links to Pooled Variance Estimator and Two-Sample T-Interval for Means
Practice and application:
- Answer Remember/Understand questions
- Answer Apply/Analyze questions
- Attempt Evaluate/Create questions
- Rework both examples (English vs Literature students, and the second numerical example) without looking at notes
Deep dive study:
- Focus on Challenging Concepts (especially the unbiasedness proof)
- Extend atomic notes for Pooled Variance Estimator and Two-Sample T-Interval for Means
- Review the chi-square independence property and how it relates to pooled variance distribution
Verification and integration:
- Cross-reference notes with textbook sections on two-sample inference
- Identify remaining gaps in understanding the T-statistic construction
- Link this lecture to STA258 Lecture 08 (single-sample CI) and future lectures on hypothesis testing