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Regression

Simple Linear Regression — Predict GPA from Study Hours

Fit OLS regression line to predict GPA from study hours. Slope, intercept, R², SE, and ANOVA table.

📊 b₁ = 0.063, R² = .970, F(1,6) = 194.1
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📈 6 · Simple Linear Regression

Predict GPA from Study Hours
Regression
Research Question: Using the same 8 students, build a regression model to predict GPA from study hours.
OLS Regression Coefficients
b 1 = ∑ ( x i − x ¯ ) ( y i − y ¯ ) ∑ ( x i − x ¯ ) 2 = 17.30 166 = 0.1042 b 0 = y ¯ − b 1 x ¯ = 3.40 − 0.1042 × 15 = 3.40 − 1.563 = 1.837 y ^ = 1.837 + 0.1042 x
R², F, SE
S S R e s = ∑ ( y i − y ^ i ) 2 = 0.040 , S S T o t = 1.843 R 2 = 1 − S S R e s S S T o t = 1 − 0.040 1.843 = 0.978 F ( 1 , 6 ) = R 2 / ( p − 1 ) ( 1 − R 2 ) / ( n − p ) = 0.978 / 1 0.022 / 6 = 0.978 0.00367 = 267.0 S E b 1 = S S R e s / ( n − 2 ) ∑ ( x − x ¯ ) 2 = 0.040 / 6 166 = 0.0000402 = 0.00634
CoefficientBSEtp95% CI
b₀ (Intercept)1.8370.10717.17<.001[1.575, 2.099]
b₁ (Hours)0.10420.0063416.44<.001[0.0887, 0.1197]
0.978
R²
267.0
F(1,6)
<.001
p
0.1042
β (slope)
✅ Every additional study hour increases predicted GPA by 0.104. The model explains 97.8% of variance.
APA-7
Simple linear regression was used to predict GPA from weekly study hours. The model was statistically significant, F(1, 6) = 267.0, p < .001, R² = .978. Study hours significantly predicted GPA (b = 0.10, SE = 0.006, t(6) = 16.44, p < .001, 95% CI [0.09, 0.12]).

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