1. **Problem Statement:** Calculate the regression coefficients of $Y$ on $X$ and find the regression equation for the given data: | X | 1 | 3 | 4 | 6 | 8 | 9 | 11 | 14 | |---|---|---|---|---|---|---|----|----| | Y | 1 | 2 | 2 | 4 | 5 | 7 | 8 | 10 | 2. **Formula for Regression Coefficients:** The regression line of $Y$ on $X$ is given by: $$\hat{Y} = a + bX$$ where $$b = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{\sum (X_i - \bar{X})^2}$$ and $$a = \bar{Y} - b\bar{X}$$ 3. **Calculate Means:** $$\bar{X} = \frac{1+3+4+6+8+9+11+14}{8} = \frac{56}{8} = 7$$ $$\bar{Y} = \frac{1+2+2+4+5+7+8+10}{8} = \frac{39}{8} = 4.875$$ 4. **Calculate the sums for numerator and denominator of $b$:** Calculate each $(X_i - \bar{X})$, $(Y_i - \bar{Y})$, their product, and $(X_i - \bar{X})^2$: | $X_i$ | $Y_i$ | $X_i - \bar{X}$ | $Y_i - \bar{Y}$ | $(X_i - \bar{X})(Y_i - \bar{Y})$ | $(X_i - \bar{X})^2$ | |-------|-------|-----------------|-----------------|---------------------------------|---------------------| | 1 | 1 | $1 - 7 = -6$ | $1 - 4.875 = -3.875$ | $(-6)(-3.875) = 23.25$ | $(-6)^2 = 36$ | | 3 | 2 | $3 - 7 = -4$ | $2 - 4.875 = -2.875$ | $(-4)(-2.875) = 11.5$ | $(-4)^2 = 16$ | | 4 | 2 | $4 - 7 = -3$ | $2 - 4.875 = -2.875$ | $(-3)(-2.875) = 8.625$ | $(-3)^2 = 9$ | | 6 | 4 | $6 - 7 = -1$ | $4 - 4.875 = -0.875$ | $(-1)(-0.875) = 0.875$ | $(-1)^2 = 1$ | | 8 | 5 | $8 - 7 = 1$ | $5 - 4.875 = 0.125$ | $(1)(0.125) = 0.125$ | $(1)^2 = 1$ | | 9 | 7 | $9 - 7 = 2$ | $7 - 4.875 = 2.125$ | $(2)(2.125) = 4.25$ | $(2)^2 = 4$ | | 11 | 8 | $11 - 7 = 4$ | $8 - 4.875 = 3.125$ | $(4)(3.125) = 12.5$ | $(4)^2 = 16$ | | 14 | 10 | $14 - 7 = 7$ | $10 - 4.875 = 5.125$ | $(7)(5.125) = 35.875$ | $(7)^2 = 49$ | Sum these values: $$\sum (X_i - \bar{X})(Y_i - \bar{Y}) = 23.25 + 11.5 + 8.625 + 0.875 + 0.125 + 4.25 + 12.5 + 35.875 = 96.5$$ $$\sum (X_i - \bar{X})^2 = 36 + 16 + 9 + 1 + 1 + 4 + 16 + 49 = 132$$ 5. **Calculate slope $b$:** $$b = \frac{96.5}{132} \approx 0.7311$$ 6. **Calculate intercept $a$:** $$a = \bar{Y} - b\bar{X} = 4.875 - 0.7311 \times 7 = 4.875 - 5.1177 = -0.2427$$ 7. **Regression Equation:** $$\hat{Y} = -0.2427 + 0.7311X$$ **Final answer:** The regression equation of $Y$ on $X$ is $$\hat{Y} = -0.243 + 0.731X$$ (rounded to three decimals).