This page gives you a sample Excel-style Multiple Linear Regression (MLR) practice set, similar to what you might see on the final exam. It matches the Chapter 12 Excel ICE idea:
On the actual exam, you will read and interpret output. You will not be asked to derive formulas from scratch.
Each row is one student. Y = GPA. Predictors:
When you run the regression on this dataset, you should get output approximately like this:
| R | R Square | Adjusted R Square | Standard Error | Observations |
|---|---|---|---|---|
| 0.89 | 0.79 | 0.76 | 0.107 | 25 |
| Source | df | SS | MS | F | Significance F |
|---|---|---|---|---|---|
| Regression | 3 | 0.890 | 0.297 | 25.9 | 0.0000003 |
| Error | 21 | 0.241 | 0.0115 | ||
| Total | 24 | 1.131 |
| Predictor | Coefficient | Standard Error | t Stat | p-value | Lower 95% | Upper 95% |
|---|---|---|---|---|---|---|
| Intercept | 1.140 | 0.129 | 8.81 | 0.0000 | 0.871 | 1.409 |
| HWhrs | 0.047 | 0.008 | 6.06 | 0.0000 | 0.031 | 0.063 |
| Tutor | 0.120 | 0.019 | 6.45 | 0.0000 | 0.082 | 0.159 |
| Sleep | 0.032 | 0.017 | 1.93 | 0.068 | -0.003 | 0.066 |
| Predictor | VIF |
|---|---|
| HWhrs | 1.03 |
| Tutor | 1.04 |
| Sleep | 1.06 |
Small VIF values (≈1) mean there is no serious multicollinearity here.
Write the fitted regression equation for GPA in terms of HWhrs, Tutor, and Sleep using the coefficients above.
At α = 0.05, does the regression model (all three predictors together) significantly
explain variation in GPA? Use the ANOVA table and “Significance F”.
True/False: “The model is statistically significant at α = 0.05.”
True. The Significance F is about 0.0000003, which is much smaller than 0.05. So we reject H₀: β₁ = β₂ = β₃ = 0 and conclude that, taken together, the predictors significantly explain GPA.
At α = 0.05, which predictors are statistically significant (have slopes different from 0)?
Look at p-values for each slope:
So HWhrs and Tutor are significant predictors at α = 0.05. Sleep is borderline (you could mention this in words).
Interpret R² = 0.79 and Adjusted R² = 0.76 in plain English, for this example.
R² = 0.79 means that about 79% of the variation in GPA among these students is explained by the regression model using HWhrs, Tutor, and Sleep.
Adjusted R² = 0.76 is slightly smaller because it penalizes us for adding predictors. It is a better measure for comparing models with different numbers of X’s.
Use the fitted equation to predict the GPA of a student with:
Compute:
Ŷ ≈ 1.140 + 0.282 + 0.240 + 0.224 ≈ 1.89 (predicted GPA ≈ 1.89).
On the exam, show at least one line of work or use Excel with a clearly labeled formula.
Give a plain-language interpretation of the HWhrs slope (0.047), assuming the model is appropriate.
For each additional hour per week spent on homework, the model predicts that GPA will increase by about 0.047 points, on average, holding Tutor and Sleep constant.
Look at the VIF values: 1.03 (HWhrs), 1.04 (Tutor), 1.06 (Sleep).
True/False: “This model has a serious multicollinearity problem.”
False. VIF values close to 1 mean the predictors are not strongly correlated with each other. Common “worry” thresholds are VIF > 5 or VIF > 10. Here, the VIFs are small, so there is no serious multicollinearity problem.
The 95% CI for the Tutor slope is given as (0.082, 0.159).
True/False: “At 95% confidence, the true Tutor effect could be 0 (no effect).”
False. The 95% CI (0.082, 0.159) does not include 0, so at the 5% significance level we conclude the Tutor slope is different from 0 (Tutor is a significant predictor).
Explain how the global F-test and the t-tests for individual slopes are related but not the same.
The global F-test checks whether the model with all predictors explains Y better than a model with no predictors (H₀: all slopes = 0). It answers “Is the regression useful at all?”
The t-tests for individual slopes ask, one predictor at a time, whether that slope is 0 given that the others are in the model. You can have a significant F-test even if one of the individual predictors is not significant.
On the actual exam, the numbers and variables may change, but you should be able to: