1. **Stating the problem:** We analyze the data from Nehema's thesis to answer questions about risk, variables, hypotheses, selection procedures, correlations, model fit, variable choice, elasticities, and prediction. 2. **Which variable is less risky?** Risk here relates to variability (standard deviation). From Table 4.1, Math Score has the lowest standard deviation $0.91$, so Math Score is less risky. 3. **Identify the output (dependent) variable:** The model predicts an outcome using Math, Reading, Social Studies, and Gender. The dependent variable is likely Science Score (Percent) as it is not an input in the regression but appears in correlations and descriptive stats. 4. **Set and explain the null hypothesis ($H_0$):** The null hypothesis tested is that the regression coefficients for the predictors are zero, meaning no effect on Science Score. $$H_0: \beta_{Math} = \beta_{Reading} = \beta_{SocialStudies} = \beta_{Gender} = 0$$ This means none of the independent variables significantly predict Science Score. 5. **Which selection procedure was used and why?** The model includes all variables simultaneously, suggesting a multiple linear regression with all predictors included to assess their individual contributions. 6. **Comment on correlation analysis (Table 4.2):** - Science Score correlates strongly with Math ($0.728$), Reading ($0.694$), and Social Studies ($0.639$), indicating these subjects relate positively to Science performance. - Gender correlations are low ($<0.12$), suggesting gender has little linear association with scores. - Implication: Math and Reading are important predictors; gender less so. 7. **Goodness-of-fit and significance (Table 4.3):** - Multiple R is $0.984$, indicating a very strong linear relationship. - Adjusted $R^2 = 0.964$ means 96.4% of variance in Science Score is explained by the model. - Sum of Squares Total $= 5436.957$, Regression $= 5265.097$ shows most variance explained. - F-value and residuals missing but high $R^2$ and Multiple R imply overall significance. - Individual coefficients: Math and Reading have significant coefficients (confidence intervals do not include zero), Social Studies less so, Gender moderately significant. 8. **If resources limited, which input variable to choose?** Math Score, because it has the highest standardized coefficient ($0.601$) and significant positive effect on Science Score. 9. **Elasticities and type:** Elasticity here is the standardized coefficient showing % change in Science Score per 1 SD change in predictor. - Math Score elasticity: $0.601$ (positive, elastic) - Reading Score elasticity: $-0.615$ (negative, elastic) - Social Studies: $0.054$ (near zero, inelastic) - Gender: $0.321$ (moderate elasticity) 10. **Compute predicted Science Score for Harriet Owenga:** Using regression equation: $$\hat{Y} = 44.720 + 2.968 \times Math - 0.679 \times Reading + 0.050 \times SocialStudies + 1.245 \times Gender$$ Assuming gender unknown, omit or set to 0 (male) or 1 (female). Since gender not given, assume 0. Plug in scores: $$\hat{Y} = 44.720 + 2.968 \times 50 - 0.679 \times 95 + 0.050 \times 70 + 1.245 \times 0$$ Calculate stepwise: $$2.968 \times 50 = 148.4$$ $$-0.679 \times 95 = -64.505$$ $$0.050 \times 70 = 3.5$$ Sum: $$44.720 + 148.4 - 64.505 + 3.5 = 132.115$$ So predicted Science Score is approximately $132.12$ (which may indicate a scale or data issue, but this is the direct calculation).