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Multiple Reg.

Multiple Regression — Predict Exam Score from Study & Sleep Hours

Two-predictor OLS regression. Normal equations, β weights (b₁=3.39, b₂=2.57), R²=.973, F(2,7)=125.2.

📊 R² = .973, F(2,7) = 125.2, p < .001
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📊 14 · Multiple Linear Regression

Predict Exam Score from Study Hours + Sleep Hours
Multiple Regression
Research Question: Can weekly study hours (X₁) and nightly sleep hours (X₂) together predict exam scores (Y) in a sample of 10 students?
StudentX₁ (Study hrs)X₂ (Sleep hrs)Y (Score)Ŷe = Y−Ŷ
1465855.32.7
2676464.7−0.7
3887274.0−2.0
4555556.1−1.1
5987877.40.6
6777068.11.9
71098583.41.6
8365251.90.1
9866868.9−0.9
10686567.2−2.2
x̄6.67.066.7
OLS Normal Equations (2 predictors)
S x 1 x 1 = 44.4 , S x 2 x 2 = 14.0 , S x 1 x 2 = 18.0 S y x 1 = 196.8 , S y x 2 = 97.0 b 1 = S y x 1 S x 2 x 2 − S y x 2 S x 1 x 2 S x 1 x 1 S x 2 x 2 − S x 1 x 2 2 = 196.8 × 14 − 97.0 × 18 44.4 × 14 − 18 2 = 1009.2 297.6 = 3.39 b 2 = S y x 2 S x 1 x 1 − S y x 1 S x 1 x 2 297.6 = 764.4 297.6 = 2.57 b 0 = Y ¯ − b 1 X ¯ 1 − b 2 X ¯ 2 = 66.7 − 3.39 ( 6.6 ) − 2.57 ( 7.0 ) = 26.34 Y ^ = 26.34 + 3.39 X 1 + 2.57 X 2
PredictorbSEtp95% CIβ (std)
Intercept26.344.186.30<.001[17.1, 35.6]—
Study Hours (X₁)3.390.418.27<.001[2.4, 4.4].74
Sleep Hours (X₂)2.570.743.47.010[0.9, 4.2].32
R², Adjusted R², F
S S t o t = 942.1 , S S r e s = 25.6 , S S r e g = 916.5 R 2 = 916.5 942.1 = .973 , R a d j 2 = 1 − 25.6 / 7 942.1 / 9 = .965 F ( 2 , 7 ) = 916.5 / 2 25.6 / 7 = 458.3 3.66 = 125.2 , p < .001
.973
R²
.965
Adj R²
125.2
F(2,7)
<.001
p
✅ The two-predictor model explains 97.3% of variance. Each extra study hour adds 3.39 points; each extra sleep hour adds 2.57 points.
APA-7
Multiple linear regression was used to predict exam scores from study hours and sleep hours. The overall model was statistically significant, F(2, 7) = 125.2, p < .001, R² = .973, adjusted R² = .965. Study hours (b = 3.39, SE = 0.41, β = .74, t(7) = 8.27, p < .001) and sleep hours (b = 2.57, SE = 0.74, β = .32, t(7) = 3.47, p = .010) were both significant predictors.

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