Pearson Correlation Aba2C8

1. **State the problem:** We want to test if there is a significant relationship between the final grades in Biology (X) and Chemistry (Y) for ten students using Pearson’s Product Moment Correlation Coefficient. 2. **Formulate hypotheses:** - Null hypothesis ($H_0$): There is no significant relationship between Biology and Chemistry grades. - Alternative hypothesis ($H_a$): There is a significant relationship between Biology and Chemistry grades. 3. **Level of significance:** $\alpha = 0.05$ 4. **Data and calculations:** | Biology (X) | Chemistry (Y) | $XY$ | $X^2$ | $Y^2$ | |-------------|---------------|------|-------|-------| | 95 | 85 | 8075 | 9025 | 7225 | | 85 | 95 | 8075 | 7225 | 9025 | | 80 | 70 | 5600 | 6400 | 4900 | | 75 | 65 | 4875 | 5625 | 4225 | | 60 | 70 | 4200 | 3600 | 4900 | | 88 | 88 | 7744 | 7744 | 7744 | | 90 | 88 | 7920 | 8100 | 7744 | | 92 | 90 | 8280 | 8464 | 8100 | | 75 | 90 | 6750 | 5625 | 8100 | | 75 | 75 | 5625 | 5625 | 5625 | Calculate sums: $$\sum X = 95 + 85 + 80 + 75 + 60 + 88 + 90 + 92 + 75 + 75 = 815$$ $$\sum Y = 85 + 95 + 70 + 65 + 70 + 88 + 88 + 90 + 90 + 75 = 816$$ $$\sum XY = 8075 + 8075 + 5600 + 4875 + 4200 + 7744 + 7920 + 8280 + 6750 + 5625 = 67144$$ $$\sum X^2 = 9025 + 7225 + 6400 + 5625 + 3600 + 7744 + 8100 + 8464 + 5625 + 5625 = 66433$$ $$\sum Y^2 = 7225 + 9025 + 4900 + 4225 + 4900 + 7744 + 7744 + 8100 + 8100 + 5625 = 67588$$ 5. **Number of pairs:** $N = 10$ 6. **Degrees of freedom:** $$df = N - 2 = 10 - 2 = 8$$ 7. **Compute Pearson’s r:** $$r = \frac{N \sum XY - (\sum X)(\sum Y)}{\sqrt{[N \sum X^2 - (\sum X)^2][N \sum Y^2 - (\sum Y)^2]}}$$ Substitute values: $$r = \frac{10 \times 67144 - 815 \times 816}{\sqrt{[10 \times 66433 - 815^2][10 \times 67588 - 816^2]}}$$ Calculate numerator: $$10 \times 67144 = 671440$$ $$815 \times 816 = 665040$$ $$\text{Numerator} = 671440 - 665040 = 6400$$ Calculate denominator parts: $$10 \times 66433 = 664330$$ $$815^2 = 664225$$ $$10 \times 67588 = 675880$$ $$816^2 = 665856$$ Calculate inside square root: $$664330 - 664225 = 105$$ $$675880 - 665856 = 10024$$ Calculate denominator: $$\sqrt{105 \times 10024} = \sqrt{1052520} \approx 1025.9$$ 8. **Calculate r:** $$r = \frac{6400}{1025.9} \approx 6.24$$ This value is greater than 1, which is impossible for Pearson's r. This suggests a calculation error in sums or data. **Recheck sums:** Sum $X$ and $Y$ are correct. Sum $XY$ seems high; re-add: $8075 + 8075 = 16150$ $16150 + 5600 = 21750$ $21750 + 4875 = 26625$ $26625 + 4200 = 30825$ $30825 + 7744 = 38569$ $38569 + 7920 = 46489$ $46489 + 8280 = 54769$ $54769 + 6750 = 61519$ $61519 + 5625 = 67144$ (correct) Sum $X^2$: $9025 + 7225 = 16250$ $16250 + 6400 = 22650$ $22650 + 5625 = 28275$ $28275 + 3600 = 31875$ $31875 + 7744 = 39619$ $39619 + 8100 = 47719$ $47719 + 8464 = 56183$ $56183 + 5625 = 61808$ $61808 + 5625 = 67433$ (corrected from 66433) Sum $Y^2$: $7225 + 9025 = 16250$ $16250 + 4900 = 21150$ $21150 + 4225 = 25375$ $25375 + 4900 = 30275$ $30275 + 7744 = 38019$ $38019 + 7744 = 45763$ $45763 + 8100 = 53863$ $53863 + 8100 = 61963$ $61963 + 5625 = 67588$ (correct) Recalculate denominator parts: $$10 \times 67433 = 674330$$ $$815^2 = 664225$$ $$674330 - 664225 = 10105$$ $$10 \times 67588 = 675880$$ $$816^2 = 665856$$ $$675880 - 665856 = 10024$$ Denominator: $$\sqrt{10105 \times 10024} = \sqrt{101292620} \approx 10064.4$$ Calculate r: $$r = \frac{6400}{10064.4} \approx 0.636$$ 9. **Decision:** Degrees of freedom $df = 8$, level of significance $\alpha = 0.05$. From correlation tables, critical value for $r$ at $df=8$ and $\alpha=0.05$ (two-tailed) is approximately $0.632$. Since computed $r = 0.636 > 0.632$, we reject $H_0$. 10. **Conclusion:** There is a significant positive correlation between Biology and Chemistry grades. This means as Biology grades increase, Chemistry grades tend to increase as well.