1. **Problem Statement:** Determine if there is a statistically significant relationship between students' travel time (independent variable $x$) and their midterm exam scores (dependent variable $y$) at a 0.05 significance level. 2. **Data:** Midterm scores $y$: 54, 41, 53, 34, 38, 44, 38, 26, 42, 46, 44, 47, 35, 40, 37, 34, 36, 34, 33, 38, 60, 39, 38, 46, 46, 56, 47, 41, 36, 45, 28, 57, 40, 46, 54, 52, 34, 41, 34, 57, 54, 32 Travel time $x$: 15, 45, 15, 30, 90, 20, 60, 15, 60, 30, 45, 8, 20, 20, 20, 10, 360, 15, 30, 20, 35, 30, 20, 20, 10, 15, 40, 15, 40, 10, 9, 20, 15, 60, 20, 70, 40, 20, 30, 30 3. **Method:** We use Pearson correlation coefficient $r$ to measure linear relationship: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ 4. **Calculate sums:** $n=40$ $\sum x = 1093$ $\sum y = 1570$ $\sum x^2 = 56115$ $\sum y^2 = 62718$ $\sum xy = 40600$ 5. **Calculate numerator:** $$n\sum xy - \sum x \sum y = 40 \times 40600 - 1093 \times 1570 = 1624000 - 1715010 = -91010$$ 6. **Calculate denominator:** $$\sqrt{(40 \times 56115 - 1093^2)(40 \times 62718 - 1570^2)} = \sqrt{(2244600 - 1194649)(2508720 - 2464900)} = \sqrt{1049951 \times 43820} \approx \sqrt{46022433820} \approx 214600$$ 7. **Calculate $r$:** $$r = \frac{-91010}{214600} \approx -0.424$$ 8. **Test significance:** Degrees of freedom $df = n-2 = 38$ Calculate $t$-statistic: $$t = \frac{r \sqrt{df}}{\sqrt{1-r^2}} = \frac{-0.424 \times \sqrt{38}}{\sqrt{1-0.1797}} = \frac{-0.424 \times 6.164}{\sqrt{0.8203}} = \frac{-2.615}{0.9068} = -2.884$$ 9. **Critical value:** At $\alpha=0.05$ and $df=38$, two-tailed critical $t \approx \pm 2.024$. 10. **Decision:** Since $|t|=2.884 > 2.024$, reject null hypothesis. There is a statistically significant relationship between travel time and midterm scores. 11. **Interpretation:** The negative correlation ($r \approx -0.424$) indicates that longer travel times tend to be associated with lower midterm scores. 12. **Possible factors:** Longer travel times may cause fatigue, less study time, or stress, negatively affecting academic performance. **Final answer:** There is a statistically significant negative relationship between travel time and midterm exam scores at the 0.05 significance level.