1. **State the problem:** We have data for 10 employees showing the number of training sessions ($x$) and the time taken to complete a task ($y$). We need to find the correlation coefficient $r$, comment on the relationship, find the regression line equation, and estimate the time for 6 sessions. 2. **Data:** $$x = [3,4,5,3,7,7,8,9,9,8]$$ $$y = [10,15,14,12,7,12,6,5,6,4]$$ 3. **Formulas:** - Correlation coefficient: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ - Regression line of $y$ on $x$: $$y = a + bx$$ where $$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$$ $$a = \bar{y} - b\bar{x}$$ 4. **Calculate sums:** $$\sum x = 3+4+5+3+7+7+8+9+9+8 = 63$$ $$\sum y = 10+15+14+12+7+12+6+5+6+4 = 91$$ $$\sum x^2 = 3^2+4^2+5^2+3^2+7^2+7^2+8^2+9^2+9^2+8^2 = 3^2+4^2+5^2+3^2+7^2+7^2+8^2+9^2+9^2+8^2 = 9+16+25+9+49+49+64+81+81+64 = 447$$ $$\sum y^2 = 10^2+15^2+14^2+12^2+7^2+12^2+6^2+5^2+6^2+4^2 = 100+225+196+144+49+144+36+25+36+16 = 971$$ $$\sum xy = 3\times10 + 4\times15 + 5\times14 + 3\times12 + 7\times7 + 7\times12 + 8\times6 + 9\times5 + 9\times6 + 8\times4 = 30 + 60 + 70 + 36 + 49 + 84 + 48 + 45 + 54 + 32 = 508$$ 5. **Calculate $r$ numerator and denominator:** $$n = 10$$ $$n\sum xy - \sum x \sum y = 10 \times 508 - 63 \times 91 = 5080 - 5733 = -653$$ $$n\sum x^2 - (\sum x)^2 = 10 \times 447 - 63^2 = 4470 - 3969 = 501$$ $$n\sum y^2 - (\sum y)^2 = 10 \times 971 - 91^2 = 9710 - 8281 = 1429$$ 6. **Calculate $r$:** $$r = \frac{-653}{\sqrt{501 \times 1429}} = \frac{-653}{\sqrt{716,929}} = \frac{-653}{846.13} \approx -0.771$$ 7. **Interpretation:** The correlation coefficient $r \approx -0.771$ indicates a strong negative linear relationship between the number of training sessions and the time taken to complete the task. As training sessions increase, time taken tends to decrease. 8. **Calculate slope $b$ of regression line:** $$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{-653}{501} \approx -1.303$$ 9. **Calculate means:** $$\bar{x} = \frac{63}{10} = 6.3$$ $$\bar{y} = \frac{91}{10} = 9.1$$ 10. **Calculate intercept $a$:** $$a = \bar{y} - b\bar{x} = 9.1 - (-1.303)(6.3) = 9.1 + 8.204 = 17.304$$ 11. **Regression line equation:** $$y = 17.304 - 1.303x$$ 12. **Estimate time for 6 sessions:** $$y = 17.304 - 1.303 \times 6 = 17.304 - 7.818 = 9.486$$ **Final answers:** - Correlation coefficient $r \approx -0.771$ - Regression line: $y = 17.304 - 1.303x$ - Estimated time for 6 sessions: approximately 9.49 minutes