📘 Chapter 11: Regression and Correlation

🎮 Interactive Apps (HTML/JS)

• ICE — Excel Data Analysis (ToolPak)
• ICE — Excel Functions (SLOPE/INTERCEPT/RSQ/STEYX)
• 11.1 – Empirical Model Intro
• 11.2 – 11.6 Simple Linear Regression — Original
• 11.3 – Least Squares Properties — Original
• 11.7 – Residual Plot & R² — Original
• 11.8 – Correlation & Inference (FYI)
• 11.9 – Transformed Variables (FYI)
• 11.10 – Logistic Regression Model (FYI)

📽️ PPT Slides

• Download Chapter 11 Slides (ch11.pptx)

📝 Quiz

• Quiz — ToolPak (10 T/F)
• Quiz — Excel Functions (15 T/F)

Open in a new tab. Scores show on-page; use the Reset button to retake.

📚 Homework

• Homework — Regression (use ToolPak or Functions)

This assignment uses the 23-student Sleep vs Grade dataset and lets you choose either approach: (1) Excel Data Analysis → Regression (ToolPak) or (2) Excel functions (SLOPE/INTERCEPT/RSQ/STEYX). Clear steps + a clean write-up template are on the homework page.

💬 Excel

📢 Student Q&A

• Q: What does the slope β₁ mean?
  A: The expected change in Y for a one-unit increase in X (units: Y-units per X-unit).
• Q: What does the intercept β₀ mean?
  A: The model’s expected Y when X = 0. If X=0 is outside your observed range, treat it as algebra (don’t over-interpret).
• Q: What’s the difference among Multiple R, R², and Adjusted R²?
  A: Multiple R is |r| (strength of linear association). R² is the percent of Y variation explained by the line (SSR/SST). Adjusted R² adds a small penalty for extra predictors (same as R² in simple one-X models up to rounding).
• Q: What does “Standard Error” (in Regression Statistics) mean?
  A: It’s √MSE—the typical vertical miss of the line in Y-units (≈ Excel’s STEYX).
• Q: What is the ANOVA F-test testing?
  A: H₀: slope β₁ = 0 (no straight-line signal). A small “Significance F” means reject H₀ → the line explains Y better than a flat mean.
• Q: Why is F = t(β₁)² in simple regression?
  A: With one X, the model test and the slope test are algebraically the same; the F-stat equals the square of the slope’s t-stat.
• Q: What does the p-value for the slope mean?
  A: The chance (if β₁ = 0 were true) of observing a slope at least this extreme. Small p → strong evidence β₁ ≠ 0.
• Q: Is a high R² proof of causation?
  A: No. R² shows fit, not cause. You still need theory, design, and checks for confounders.
• Q: When is a model “good”?
  A: High R², small Standard Error (vs Y’s scale), small p for slope & F, and residuals look random (no curve or pattern, no extreme outliers).
• Q: Should I force the intercept to zero?
  A: Usually no. Forcing β₀ = 0 can bias β₁ unless you have a solid theoretical reason (and data include X near 0).
• Q: What’s the difference between confidence and prediction intervals?
  A: A 95% CI is for the mean ŷ at a given X; a 95% PI is for an individual new Y (wider). Use PI for single-student forecasts.
• Q: What is extrapolation and why is it risky?
  A: Predicting beyond your X-range. The line may not hold outside the data—warn students when X₀ is out-of-range.
• Q: What if residuals curve?
  A: The relationship may be nonlinear. Consider transforming variables or using a curved/multiple-X model (beyond SLR).
• Q: What if residual spread grows with X (fan shape)?
  A: That’s heteroskedasticity. In SLR class: flag it and be cautious with inference; in advanced courses use robust SEs or transform variables.
• Q: What about outliers?
  A: Check residuals and leverage. Outliers can swing the line; verify data entry, then decide whether to keep with explanation or run sensitivity checks.
• Q: Negative slope—what does it mean?
  A: As X increases, Y tends to decrease by |β₁| per X-unit (units still Y/X). Direction comes from the sign.
• Q: STEYX vs SE of slope vs SE of intercept?
  A: STEYX ≈ √MSE (typical vertical error). SE(β₁) scales like √(MSE/Sxx) and controls the t-test for slope. SE(β₀) combines MSE with X-range and n.
• Q: How many observations do I need?
  A: More is better. As a classroom rule of thumb, aim for ≥ 20–25 to stabilize estimates and see patterns in residuals.
• Q: ToolPak vs Functions—when to use which?
  A: ToolPak gives an instant Summary Output (stats + ANOVA + residuals). Functions teach what each piece means and let you build custom sheets.
• Q: What are the df values in ANOVA?
  A: Regression df = #predictors (=1). Residual df = n − (#predictors + 1) (= n − 2). Total df = n − 1.
• Q: Does changing units change results?
  A: Slope rescales (e.g., hours → minutes multiplies β₁), but fitted ŷ stays correct if you use consistent units for X and β.
• Q: Can I include a yes/no (categorical) X?
  A: Yes via a dummy variable (0/1), but that’s multiple regression content (next chapter). ToolPak supports it.
• Q: How do I make a quick forecast in Excel?
  A: If β₀ is in E2 and β₁ in E3, and your chosen X is in H2, use = $E$2 + $E$3 * H2. (Or =FORECAST.LINEAR(H2, Y-range, X-range).)
• Q: Why can R² be “good” but predictions still be off?
  A: R² is global fit. Individual predictions still have noise ≈ Standard Error; use PIs to reflect that uncertainty.
• Q: My correlation is strong but p-value isn’t tiny—why?
  A: Small samples make p-values larger. t depends on both r and n.
• Q: Where do the sums of squares names come from?
  A: SST total = Σ(y−ȳ)², SSR regression = Σ(ŷ−ȳ)², SSE error = Σ(y−ŷ)², and SST = SSR + SSE.