91Ó°ÊÓ

To obtain the sums of squares and cross-products, we need the sums of \(x, y, x^2, xy,\) and \(y^2\). We will start by listing the given data points and calculating the required sums: $$ \begin{array}{c|rrrrrrr} x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 1 & 1 & 3 & 5 & 5 \end{array} $$ Now, we will calculate the required sums: - \(\sum x = -2 - 1 + 0 + 1 + 2 = 0\) - \(\sum y = 1 + 1 + 3 + 5 + 5 = 15\) - \(\sum x^2 = (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 10\) - \(\sum xy = (-2)(1) + (-1)(1) + 0(3) + 1(5) + 2(5) = 12\) - \(\sum y^2 = 1^2 + 1^2 + 3^2 + 5^2 + 5^2 = 61\)

Now, we will calculate the sums of squares and cross-products using the formulas: - \(S_{xx} = \sum x^2 - \frac{(\sum x)^2}{n} = 10 - \frac{0^2}{5} = 10\) - \(S_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n} = 12 - \frac{0\cdot15}{5} = 12\) - \(S_{yy} = \sum y^2 - \frac{(\sum y)^2}{n} = 61 - \frac{15^2}{5} = 16\)

Next, we will find the slope (m) and intercept (b) of the least-squares line using the formulas: - \(m = \frac{S_{xy}}{S_{xx}} = \frac{12}{10} = 1.2\) - \(b = \bar{y} - m\bar{x} = \frac{\sum y}{n} - m\frac{\sum x}{n} = \frac{15}{5} - 1.2\frac{0}{5} = 3\) Thus, the least-squares line is given by the equation: $$y = 1.2x + 3$$

Now, we can plot the given points and the least-squares line found above on a graph. After doing so, visually inspect whether the line appears to be a good fit for the data points. This can be subjective but typically, if the line goes through or reasonably close to most of the data points, it can be considered a good fit.

To construct the ANOVA table, we need the following values: - SSR (Sum of Squares Regression): \(SSR = m^2 \cdot S_{xx} = 1.2^2 \cdot 10 = 14.4\) - SSE (Sum of Squares Error): \(SSE = S_{yy} - m^2 \cdot S_{xx} = 16 - 14.4 = 1.6\) - MST (Mean Square Total): \(MST = \frac{S_{yy}}{n - 1} = \frac{16}{5 - 1} = 4\) - MSR (Mean Square Regression): \(MSR = \frac{SSR}{1} = 14.4\) - MSE (Mean Square Error): \(MSE = \frac{SSE}{n - 2} = \frac{1.6}{5 - 2} = 0.533\) We can now create the ANOVA table with these values: $$ \begin{array}{c|c|c|c|c} \text{Source} & \text{SS} & \text{df} & \text{MS} & \text{F} \\ \hline \text{Regression} & 14.4 & 1 & 14.4 & 27 \\ \text{Error} & 1.6 & 3 & 0.533 & - \\ \hline \text{Total} & 16 & 4 & 4 & - \end{array} $$ The ANOVA table is now complete.