⭐ The big idea: hypothesis testing asks "is this number really different from zero, or is it just random noise?" We use the t-statistic — if it's bigger than 2 (or smaller than −2), the predictor is significant and worth keeping.
🟦 The Setup (advertising data)
We predict sales using three advertising spends: TV, radio, and newspaper. The table gives each predictor's coefficient and standard deviation.
| Predictor | Coefficient | Std. Dev. | t-stat | Significant? |
|---|---|---|---|---|
| Intercept | 2.939 | 0.3119 | 9.42 | ✅ yes |
| TV | 0.046 | 0.0014 | 32.86 | ✅ yes |
| radio | 0.189 | 0.0086 | 21.98 | ✅ yes |
| newspaper | −0.001 | 0.0059 | −0.17 | ❌ no |
⚖️ The rule of thumb: t-stat = coefficient ÷ standard deviation. If |t| > 2, the predictor is significant (green). If |t| < 2, it's not (red) — it might just be noise.
🎨 How to read this: LEFT — the fitted line y = 0 + 2x through the 5 points (this is the dataset for Questions 2–5). RIGHT — the t-statistics for Question 1: TV, radio, and intercept are way past the ±2 line (green = significant), but newspaper (−0.17) barely moves off zero (red = not significant).
1️⃣ Question 1 — What conclusions at 95% confidence?
From the table, what conclusions can be drawn at a 95% confidence level? (select all that apply)
✅ The intercept is significant (reject β₀=0) because its coefficient > 2
✅ TV is significant because |t-stat| > 2
❌ We fail to reject β_radio=0, so radio can be removed
✅ We reject β_radio=0 — radio does affect sales
✅ Newspaper 95% CI is roughly [−0.0128, 0.0108]
✅ Answer: 1, 2, 4, and 5 are true. (Option 3 is false.)
Why each one
1
Intercept: coefficient 2.939 > 2, so it's significant. We reject β₀=0. ✅
2
TV: t = 0.046/0.0014 = 32.86, way > 2. Significant. ✅
3
radio: t = 0.189/0.0086 = 21.98, way > 2. So we DO reject β_radio=0 — radio does affect sales. So option 3 (which says "fail to reject") is FALSE, and option 4 is TRUE. ✅
4
newspaper: t = −0.001/0.0059 = −0.17, tiny. Not significant. Its 95% CI = coefficient ± 2×SD = −0.001 ± 2(0.0059) = [−0.0128, 0.0108]. ✅
🎯 Darts analogy: the t-statistic is like how far your dart lands from the bullseye (zero). If it lands far enough (|t| > 2), you're confident it's a real hit, not a fluke. TV and radio are bullseyes; newspaper barely left the board.
🟦 The Setup for Questions 2–5
Nature follows the model Y = β₀ + β₁X + ε, where ε is Gaussian noise N(0, σ²). We have 5 points:
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | −3.5 | −3 | 0 | 3 | 3.5 |
📈 Perfectly symmetric: notice the points are symmetric around the origin — that's why the intercept β₀ comes out to 0.
2️⃣ Question 2 — What is β̂₁?
What is β̂₁ (the slope)? (whole number)
✅ Answer: 2
1
The slope is "how much y rises per 1 unit of x".
2
From x=−2 to x=2, y goes from −3.5 to 3.5 — that's 7 up over 4 across = 1.75... but wait, let's use the formula.
3
β₁ = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)². With x̄=0 and ȳ=0, this simplifies to Σ(x·y)/Σ(x²).
4
Σ(x·y) = (−2)(−3.5)+(−1)(−3)+(0)(0)+(1)(3)+(2)(3.5) = 7+3+0+3+7 = 20. Σ(x²) = 4+1+0+1+4 = 10. So β₁ = 20/10 = 2.
⛰️ Slope analogy: the line y = 2x rises 2 units for every 1 unit across. Steep and clean.
3️⃣ Question 3 — What is β̂₀?
What is β̂₀ (the intercept)? (whole number)
✅ Answer: 0
1
The intercept is where the line crosses the y-axis (at x=0).
2
Formula: β₀ = ȳ − β₁·x̄ = 0 − 2·0 = 0.
3
Because the data is symmetric around the origin, the line passes right through (0,0).
🏠 Starting point: β₀ is where the line starts when x=0. Here it starts exactly at zero.
4️⃣ Question 4 — What is the RSS?
Compute the RSS of the linear model. (real number, one decimal)
✅ Answer: 2.5
1
RSS = Residual Sum of Squares = add up (y − ŷ)² for every point.
2
With ŷ = 2x, the residuals are:
0.5, −1, 0, 1, −0.5.3
Square each:
0.25 + 1 + 0 + 1 + 0.25 = 2.5.📏 How far off: RSS measures the total "miss distance" of the line. Smaller = better fit. Here it's 2.5.
5️⃣ Question 5 — What is the RSE?
Compute the RSE of the model. (real number, one decimal)
✅ Answer: 0.9
1
RSE = Residual Standard Error = √(RSS / (n − 2)).
2
n = 5 points, so n−2 = 3.
3
RSE = √(2.5 / 3) = √0.8333 = 0.9129 ≈ 0.9.
📏 Average miss: RSE is like the "average" distance points sit from the line (after accounting for the 2 parameters we estimated). About 0.9 units off.
🎮 Play with it yourself
📋 Quick Answer Sheet
| # | Question | Answer |
|---|---|---|
| 1 | Conclusions at 95% (select all) | 1, 2, 4, 5 (intercept, TV, radio significant; newspaper not) |
| 2 | β̂₁ (slope) | 2 |
| 3 | β̂₀ (intercept) | 0 |
| 4 | RSS | 2.5 |
| 5 | RSE | 0.9 |
🧠 Checklist (did you get it?)
- ✅ t-stat = coefficient ÷ standard deviation.
- ✅ |t| > 2 → significant (reject null, keep predictor).
- ✅ |t| < 2 → not significant (might be noise).
- ✅ 95% CI ≈ coefficient ± 2×SD.
- ✅ β₁ = Σ(x−x̄)(y−ȳ) / Σ(x−x̄)²; β₀ = ȳ − β₁x̄.
- ✅ RSS = Σ(y−ŷ)²; RSE = √(RSS/(n−2)).