Case File — Passing & points

Phase 1 of 4

Frame the problem

Before any maths: is this a regression job, and what plays the role of X and Y?

Is this an SLR job?

1 · Is this a simple linear regression problem?

2 · Which variable is the response, Y?

3 · So which is the predictor, X?

Phase 2 of 4

You decided: SLR, with Y = points and X = passing yards. Choose each move yourself.

What's the play?

4 · What's your first move?

5 · Means done. What do you need before the slope?

6 · You've got the line. "How good is it?" — compute…?

Phase 3 of 4

The tedious sums are done for you. Apply the formulas and check yourself.

Slope, intercept, prediction

x̄245.62

ȳ23.00

Sxx5671.9

Sxy850.0

Compute the line, then predict a 260-yard passing offense:

β̂₁ (slope)

β̂₀ (intercept)

ŷ at 260 yds

β̂₁ = Sxy / Sxx = 850.0 / 5671.9 = 0.150 β̂₀ = ȳ − β̂₁·x̄ = 23.00 − 0.150 × 245.62 = -13.81 ŷ(260 yds) = -13.81 + 0.150 × 260 yds = 25.2 points

Fitted line: ŷ = -13.81 + 0.150 × yards. Each +1 passing yard is worth about 0.150 points.

Phase 4 of 4

The stats only matter if they answer the real question. Two calls to make.

So what do you tell them?

7 · An offense adds 10 passing yards per game. Best estimate of the extra points per game?

extra points

Δŷ = β̂₁ × 10 = 0.150 × 10 = 1.50 points

8 · Would you trust the model's prediction for a 350-yard passing game?

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You framed it, planned it, crunched it, and made the call — SLR as a problem-solving tool, not a recipe.