Test Yourself — Walks → Runs Allowed

sports_baseball

The scenario

When a pitcher throws four balls outside the strike zone, the batter gets a free base — a walk (BB). Walks put runners on for free, so the hunch is: teams whose pitchers walk more batters probably allow more runs. Let x = walks allowed and y = runs allowed, across seven MLB teams.

Team	Walks (x)	Runs allowed (y)
Braves	472	644
Mets	521	705
Phillies	498	672
Marlins	556	748
Nationals	541	731
Pirates	509	689
Cardinals	483	658

n = 7. Answers are checked with a little rounding tolerance, so don't sweat the last decimal.

aCalculate the sample means x̄ and ȳ.2 marks

Sum each column, divide by 7.

x̄

ȳ

x̄ = (472+521+498+556+541+509+483) / 7 = 3580 / 7 = 511.43 ȳ = (644+705+672+748+731+689+658) / 7 = 4847 / 7 = 692.43

The line will pass through (511.43, 692.43) — the average team in this sample.

bCalculate Sxx = Σ(xᵢ−x̄)² and Sxy = Σ(xᵢ−x̄)(yᵢ−ȳ).4 marks

Build a deviation table, then sum the last two columns.

Sxx

Sxy

Team	xᵢ−x̄	yᵢ−ȳ	(xᵢ−x̄)²	(xᵢ−x̄)(yᵢ−ȳ)
Braves	−39.43	−48.43	1554.7	1909.6
Mets	9.57	12.57	91.6	120.3
Phillies	−13.43	−20.43	180.4	274.4
Marlins	44.57	55.57	1986.5	2476.7
Nationals	29.57	38.57	874.4	1140.0
Pirates	−2.43	−3.43	5.9	8.3
Cardinals	−28.43	−34.43	808.3	978.6
Σ	≈ 0	≈ 0	5501.8	6907.9

Both large and positive — walks and runs allowed clearly move together.

cEstimate the slope β̂₁ and intercept β̂₀.3 marks

β̂₁ = Sxy / Sxx, then β̂₀ = ȳ − β̂₁ × x̄.

β̂₁

β̂₀

β̂₁ = Sxy / Sxx = 6907.9 / 5501.8 = 1.256 β̂₀ = ȳ − β̂₁ × x̄ = 692.43 − 1.256 × 511.43 = 50.07

ŷ = 50.07 + 1.256 × x

Slope: each extra walk allowed is associated with ≈ 1.256 more runs allowed, on average.

dFind the fitted value and residual for the Marlins (x = 556).2 marks

Plug x = 556 into the line. Residual = actual − fitted. (Enter the residual.)

residual e

ŷ = 50.07 + 1.256 × 556 = 748.41 e = y − ŷ = 748 − 748.41 = −0.41

A tiny residual — the model fits the Marlins almost perfectly.

eCalculate SSE and s². State the degrees of freedom.3 marks

Square all 7 residuals and sum for SSE; then s² = SSE / (n−2).

SSE

s²

Team	yᵢ	ŷᵢ	eᵢ	eᵢ²
Braves	644	643.0	+1.0	1.00
Mets	705	704.6	+0.4	0.16
Phillies	672	675.3	−3.3	10.89
Marlins	748	748.4	−0.4	0.16
Nationals	731	729.3	+1.7	2.89
Pirates	689	689.2	−0.2	0.04
Cardinals	658	656.8	+1.2	1.44
Σ			≈ 0	16.58

df = n − 2 = 7 − 2 = 5 s² = SSE / (n−2) = 16.58 / 5 = 3.32 (s = 1.82 runs)

Exam trapAlways divide by n−2 for SLR — never n or n−1.

fCalculate R² and interpret it.3 marks

TSS = Σ(yᵢ−ȳ)², then R² = 1 − SSE / TSS.

R²

TSS = Σ(yᵢ−ȳ)² = 8692.1 R² = 1 − SSE / TSS = 1 − 16.58 / 8692.1 = 0.998

Interpretation: walks allowed explain 99.8% of the variation in runs allowed across these 7 teams — an excellent fit.

gA new team, the Tigers, allowed 530 walks. Predict their runs allowed.1 mark

Substitute x = 530. Is this interpolation or extrapolation?

ŷ

ŷ = 50.07 + 1.256 × 530 = 715.75 ≈ 716 runs

x = 530 is inside the data range (472–556), so this is interpolation — a reliable prediction.

sports_baseball

Home run — 7 / 7!

You just ran a full simple linear regression by hand, start to finish. That's the whole chapter.

Your turn

The scenario

Home run — 7 / 7!