1. **Stating the problem:** We want to analyze the influence of motivation (x) on work spirit (y) using the given data. 2. **Data given:** | x | y | |----|----| | 6 | 7 | | 7 | 6 | | 8 | 7 | | 9 | 8 | | 9 | 9 | | 9 | 10 | | 9 | 8 | | 10 | 10 | | 10 | 12 | | 5 | 20 | 3. **Step 1: Calculate sums and squares for correlation and regression:** Calculate $\sum x$, $\sum y$, $\sum x^2$, $\sum y^2$, and $\sum xy$. - $\sum x = 6+7+8+9+9+9+9+10+10+5 = 82$ - $\sum y = 7+6+7+8+9+10+8+10+12+20 = 97$ - $\sum x^2 = 6^2+7^2+8^2+9^2+9^2+9^2+9^2+10^2+10^2+5^2 = 36+49+64+81+81+81+81+100+100+25 = 698$ - $\sum y^2 = 7^2+6^2+7^2+8^2+9^2+10^2+8^2+10^2+12^2+20^2 = 49+36+49+64+81+100+64+100+144+400 = 1087$ - $\sum xy = 6\times7 + 7\times6 + 8\times7 + 9\times8 + 9\times9 + 9\times10 + 9\times8 + 10\times10 + 10\times12 + 5\times20 = 42+42+56+72+81+90+72+100+120+100 = 775$ Number of data points $n=10$. 4. **Step 2: Calculate correlation coefficient $r$:** $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ Calculate numerator: $$10 \times 775 - 82 \times 97 = 7750 - 7954 = -204$$ Calculate denominator: $$\sqrt{(10 \times 698 - 82^2)(10 \times 1087 - 97^2)} = \sqrt{(6980 - 6724)(10870 - 9409)} = \sqrt{256 \times 1461} = \sqrt{373,416} \approx 611.07$$ So, $$r = \frac{-204}{611.07} \approx -0.334$$ 5. **Step 3: Test significance of $r$ using t-test:** $$t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}$$ Calculate: $$t = \frac{-0.334 \times \sqrt{8}}{\sqrt{1 - (-0.334)^2}} = \frac{-0.334 \times 2.828}{\sqrt{1 - 0.111}} = \frac{-0.945}{\sqrt{0.889}} = \frac{-0.945}{0.943} \approx -1.002$$ Degrees of freedom $df = n-2 = 8$. At $\alpha=0.05$, two-tailed critical t-value $\approx 2.306$. Since $|t|=1.002 < 2.306$, the correlation is **not significant**. 6. **Step 4: Calculate linear regression equation $y = a + bx$:** Slope $b$: $$b = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{7750 - 7954}{6980 - 6724} = \frac{-204}{256} = -0.797$$ Intercept $a$: $$a = \frac{\sum y}{n} - b \frac{\sum x}{n} = 9.7 - (-0.797) \times 8.2 = 9.7 + 6.535 = 16.235$$ So, $$y = 16.235 - 0.797x$$ 7. **Step 5: Calculate coefficient of determination $R^2$ to find influence:** $$R^2 = r^2 = (-0.334)^2 = 0.111$$ This means motivation explains about 11.1% of the variance in work spirit. --- **Final answers:** - The correlation coefficient between motivation and work spirit is approximately $-0.334$. - The correlation is not statistically significant at 5% level. - The linear regression equation is $y = 16.235 - 0.797x$. - The influence of motivation on work spirit is about 11.1%.