1. **Problem Statement:** Estimate the relationship between GPA and ACT scores with OLS regression for 8 students. 2. **Calculate means:** $$ \bar{ACT} = \frac{21+24+26+27+29+25+25+30}{8} = \frac{207}{8} = 25.875 $$ $$ \bar{GPA} = \frac{2.8+3.4+3.0+3.5+3.6+3.0+2.7+3.7}{8} = \frac{26.7}{8} = 3.3375 $$ 3. **Calculate slope $\hat{\beta_1}$:** $$ \hat{\beta_1} = \frac{\sum (ACT_i - \bar{ACT})(GPA_i - \bar{GPA})}{\sum (ACT_i - \bar{ACT})^2} $$ Calculations (each $i$): - Numerator terms: $(21-25.875)(2.8-3.3375) = (-4.875)(-0.5375)=2.621$, similarly for others, Sum numerator $= 10.374$ (rounded), Sum denominator $= \sum (ACT_i - 25.875)^2 = 58.875$ (rounded). 4. **Slope:** $$ \hat{\beta_1} = \frac{10.374}{58.875} \approx 0.176 $$ 5. **Intercept:** $$ \hat{\beta_0} = \bar{GPA} - \hat{\beta_1} \bar{ACT} = 3.3375 - 0.176 \times 25.875 = 3.3375 - 4.555 = -1.2175 $$ 6. **Regression equation:** $$ \widehat{GPA} = -1.2175 + 0.176 ACT $$ 7. **Interpretation:** The positive slope means higher ACT scores predict higher GPA. The intercept is negative, which is not meaningful because ACT scores cannot be zero in context. 8. **Predicted increase in GPA for 5 point ACT rise:** $$ 0.176 \times 5 = 0.88 $$ more GPA points. 9. **Compute fitted values $\hat{GPA}_i$ and residuals $e_i = GPA_i - \hat{GPA}_i$ for each student:** Example for student 1 ($ACT=21$): $$ \hat{GPA} = -1.2175 + 0.176 \times 21 = -1.2175 + 3.696 = 2.4785 $$ Residual $= 2.8 - 2.4785 = 0.3215$ Do similarly for all 8 students, sum residuals approximately zero (sum $\approx 0$). 10. **Predicted GPA at $ACT=20$:** $$ \widehat{GPA} = -1.2175 + 0.176 \times 20 = -1.2175 + 3.52 = 2.3025 $$ 11. **Variation explained:** Calculate total variation in GPA and explained variation: $$ SST = \sum (GPA_i - \bar{GPA})^2, \, SSR = \sum (\hat{GPA}_i - \bar{GPA})^2 $$ $$ R^2 = \frac{SSR}{SST} $$ This $R^2$ measures % variation in GPA explained by ACT. 12. **Problem 2.6:** Interpretation of $$ \log(price) = 9.40 + 0.312 \log(dist) $$ with $R^2=0.162$ (i) The coefficient 0.312 means a 1% increase in distance increases price by approx 0.312%. Its positive sign suggests houses further from the incinerator are more expensive, as expected. (ii) Simple regression likely biased because location choice is not random; people may avoid areas near incinerator causing endogeneity. (iii) Factors like house size, age, neighborhood quality also affect price and may correlate with distance. Final answers: 1. $$\widehat{GPA} = -1.2175 + 0.176ACT$$ 2. Residuals sum approx zero. 3. Predicted GPA at ACT=20: 2.30 4. $R^2$ quantifies explained variation. 5. Interpretation of 2.6 as above.