1. **State the problem:** Calculate the correlation coefficient $r$ for the given data points $(x, y)$. 2. **Formula:** The correlation coefficient $r$ is given by: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ where $n$ is the number of data points. 3. **Calculate sums:** - $n = 8$ - $\sum x = 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 68$ - $\sum y = 15.3 + 17.98 + 18.16 + 17.64 + 19.72 + 22.6 + 21.58 + 25.36 = 158.34$ - $\sum x^2 = 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 = 538$ - $\sum y^2 = 15.3^2 + 17.98^2 + 18.16^2 + 17.64^2 + 19.72^2 + 22.6^2 + 21.58^2 + 25.36^2 = 3203.67$ - $\sum xy = 5\times15.3 + 6\times17.98 + 7\times18.16 + 8\times17.64 + 9\times19.72 + 10\times22.6 + 11\times21.58 + 12\times25.36 = 1387.18$ 4. **Plug values into formula numerator:** $$n\sum xy - \sum x \sum y = 8 \times 1387.18 - 68 \times 158.34 = 11097.44 - 10767.12 = 330.32$$ 5. **Calculate denominator parts:** $$n\sum x^2 - (\sum x)^2 = 8 \times 538 - 68^2 = 4304 - 4624 = -320$$ $$n\sum y^2 - (\sum y)^2 = 8 \times 3203.67 - 158.34^2 = 25629.36 - 25068.36 = 561$$ 6. **Note:** The first denominator part is negative, which is impossible for variance. Recalculate $\sum x^2$: $$5^2=25, 6^2=36, 7^2=49, 8^2=64, 9^2=81, 10^2=100, 11^2=121, 12^2=144$$ Sum: $25+36+49+64+81+100+121+144=620$ 7. **Recalculate denominator parts:** $$n\sum x^2 - (\sum x)^2 = 8 \times 620 - 68^2 = 4960 - 4624 = 336$$ 8. **Calculate denominator:** $$\sqrt{336 \times 561} = \sqrt{188496} \approx 434.22$$ 9. **Calculate correlation coefficient:** $$r = \frac{330.32}{434.22} \approx 0.760$$ **Final answer:** The correlation coefficient rounded to three decimal places is **0.760**.