1. **Problem Statement:** Given data points $x = \{1,2,3,4,5,6\}$ and $y = \{5.6,4.6,4.5,3.7,3.2,2.7\}$, find the sums of squares and cross-products $S_{xx}$, $S_{yy}$, and $S_{xy}$. 2. **Calculate sums:** $$\sum x = 1+2+3+4+5+6 = 21$$ $$\sum y = 5.6+4.6+4.5+3.7+3.2+2.7 = 24.3$$ $$\sum x^2 = 1^2+2^2+3^2+4^2+5^2+6^2 = 91$$ $$\sum y^2 = 5.6^2+4.6^2+4.5^2+3.7^2+3.2^2+2.7^2 = 108.03$$ $$\sum xy = (1)(5.6)+(2)(4.6)+(3)(4.5)+(4)(3.7)+(5)(3.2)+(6)(2.7) = 74.8$$ 3. **Calculate sums of squares:** $$S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n} = 91 - \frac{21^2}{6} = 91 - 73.5 = 17.5$$ $$S_{yy} = \sum y^2 - \frac{(\sum y)^2}{n} = 108.03 - \frac{24.3^2}{6} = 108.03 - 98.505 = 9.525$$ $$S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n} = 74.8 - \frac{21 \times 24.3}{6} = 74.8 - 85.05 = -10.25$$ 4. **Find the least-squares line:** Slope: $$b = \frac{S_{xy}}{S_{xx}} = \frac{-10.25}{17.5} = -0.5857$$ Mean values: $$\bar{x} = \frac{21}{6} = 3.5, \quad \bar{y} = \frac{24.3}{6} = 4.05$$ Intercept: $$a = \bar{y} - b\bar{x} = 4.05 - (-0.5857)(3.5) = 4.05 + 2.05 = 6.10$$ Equation: $$\hat{y} = 6.10 - 0.5857x$$ 5. **Predict $y$ when $x=3.5$:** $$\hat{y} = 6.10 - 0.5857 \times 3.5 = 6.10 - 2.05 = 4.05$$ 6. **Calculate sums of squares for ANOVA:** $$SSR = \frac{S_{xy}^2}{S_{xx}} = \frac{(-10.25)^2}{17.5} = \frac{105.06}{17.5} = 6.003$$ $$SSE = S_{yy} - SSR = 9.525 - 6.003 = 3.522$$ $$\text{Total SS} = S_{yy} = 9.525$$ 7. **Construct ANOVA table:** \begin{tabular}{lcccc} Source & df & SS & MS & F \\ Regression & 1 & 6.003 & 6.003 & $\frac{6.003}{1.174} = 5.11$ \\ Error & 4 & 3.522 & 0.881 & \\ Total & 5 & 9.525 & & \end{tabular} where $MS_{Error} = \frac{SSE}{n-2} = \frac{3.522}{4} = 0.881$. **Final answers:** - $S_{xx} = 17.5$ - $S_{yy} = 9.525$ - $S_{xy} = -10.25$ - Least-squares line: $\hat{y} = 6.10 - 0.5857x$ - Predicted $y$ at $x=3.5$ is $4.05$ - ANOVA table as above