1. **Problem Statement:** Determine if there is a statistically significant relationship between students' travel time (in minutes) and their midterm exam scores at a 0.05 significance level. 2. **Statistical Method:** We use the Pearson correlation coefficient $r$ to measure the strength and direction of the linear relationship between travel time and midterm scores. 3. **Hypotheses:** - Null hypothesis $H_0$: There is no linear relationship between travel time and midterm scores, i.e., $\rho = 0$. - Alternative hypothesis $H_a$: There is a linear relationship, i.e., $\rho \neq 0$. 4. **Formula for Pearson correlation coefficient:** $$ r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}} $$ where $x$ is travel time, $y$ is midterm score, and $n$ is the number of paired observations. 5. **Calculate sums:** - $n = 42$ - $\sum x = 1113$ - $\sum y = 1683$ - $\sum xy = 42201$ - $\sum x^2 = 51219$ - $\sum y^2 = 69225$ 6. **Calculate numerator:** $$ n\sum xy - \sum x \sum y = 42 \times 42201 - 1113 \times 1683 = 1772442 - 1873179 = -100737 $$ 7. **Calculate denominator:** $$ \sqrt{(42 \times 51219 - 1113^2)(42 \times 69225 - 1683^2)} = \sqrt{(2151118 - 1238769)(2907450 - 2833089)} = \sqrt{912349 \times 74361} \approx \sqrt{67851209589} \approx 260488.5 $$ 8. **Calculate $r$:** $$ r = \frac{-100737}{260488.5} \approx -0.3867 $$ 9. **Test statistic for correlation:** $$ t = r \sqrt{\frac{n-2}{1-r^2}} = -0.3867 \sqrt{\frac{40}{1 - (-0.3867)^2}} = -0.3867 \sqrt{\frac{40}{1 - 0.1495}} = -0.3867 \sqrt{\frac{40}{0.8505}} = -0.3867 \times 6.86 = -2.65 $$ 10. **Degrees of freedom:** $df = n - 2 = 40$ 11. **Critical value:** For $\alpha = 0.05$ two-tailed test and $df=40$, $t_{critical} \approx \pm 2.021$ 12. **Decision:** Since $|t| = 2.65 > 2.021$, reject $H_0$. 13. **Interpretation:** There is a statistically significant negative linear relationship between travel time and midterm exam scores at the 0.05 significance level. This means as travel time increases, midterm scores tend to decrease. 14. **Possible factors:** Longer travel times may cause fatigue, reduce study time, or increase stress, negatively affecting academic performance. Other confounding variables may also influence this relationship. **Final answer:** There is a significant negative correlation ($r \approx -0.39$) between travel time and midterm exam scores, indicating travel time affects academic performance negatively.