10
Quantitative MethodsModule 10 of 11

Simple Linear Regression

5

Concepts

5

Formulas

1

Decisions

3

Quiz Questions

Key Concepts

5 concepts covered in this module.

Regression Equation

Ŷ = b0 + b1X. b0 = intercept, b1 = slope (change in Y per unit change in X).

R-squared

Coefficient of determination = proportion of Y's variation explained by X. R² = correlation². Range: 0 to 1.

Standard Error of Estimate (SEE)

Measures the average distance of actual Y values from the regression line. Lower SEE = better fit.

ANOVA

SST = SSR + SSE. Total variation = Explained + Unexplained. F = MSR/MSE tests overall significance.

Assumptions

Linearity, homoscedasticity (constant variance), independence of errors, normality of errors.

Formulas

5 essential formulas for this module.

Slope Coefficient

b1 = Cov(X,Y) / Var(X) = Σ(X-X̄)(Y-Ȳ) / Σ(X-X̄)²

Where: OLS estimator

Intercept

b0 = Ȳ - b1X̄

Where: Regression line passes through (X̄, Ȳ)

R-squared

R² = SSR/SST = 1 - SSE/SST

Where: SST = SSR + SSE

F-statistic

F = MSR/MSE = (SSR/1) / (SSE/(n-2))

Where: Tests overall significance

t-test for slope

t = b1 / SE(b1)

Where: H0: b1 = 0 (no relationship)

Decision Frameworks

1 decision frameworks to guide your analysis.

How to assess regression quality?

  • R² for goodness of fit
  • SEE for prediction accuracy
  • t-test on b<sub>1</sub> for statistical significance
  • F-test for overall model significance

Mind Map

Visual overview of how concepts connect in this module.

Simple Linear Regression
Equation
Y = b0 + b1×X + ε
b1 = Cov(X,Y)/Var(X)
b0 = Ȳ - b1×X̄
Goodness of Fit
R² = SSR/SST
SEE = √(SSE/(n-2))
Higher R² = better fit
Hypothesis Tests
t-test on b1
F-test (ANOVA)
In simple: t² = F
Assumptions
Linearity
Homoscedasticity
Independence
Normal errors
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Regression Equation

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Answer
Ŷ = b0 + b1X. b0 = intercept, b1 = slope (change in Y per unit change in X).
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