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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
$$b_1 = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2} = \frac{17.30}{166} = 0.1042$$ $$b_0 = \bar{y} - b_1\bar{x} = 3.40 - 0.1042 \times 15 = 3.40 - 1.563 = 1.837$$ $$\hat{y} = 1.837 + 0.1042x$$
R², F, SE
$$SS_{Res} = \sum(y_i - \hat{y}_i)^2 = 0.040, \quad SS_{Tot} = 1.843$$ $$R^2 = 1 - \frac{SS_{Res}}{SS_{Tot}} = 1 - \frac{0.040}{1.843} = 0.978$$ $$F(1,6) = \frac{R^2/(p-1)}{(1-R^2)/(n-p)} = \frac{0.978/1}{0.022/6} = \frac{0.978}{0.00367} = 267.0$$ $$SE_{b_1} = \sqrt{\frac{SS_{Res}/(n-2)}{\sum(x-\bar{x})^2}} = \sqrt{\frac{0.040/6}{166}} = \sqrt{0.0000402} = 0.00634$$
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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