Forming ANOVA table without observed values.

Question

Need some help in forming the ANOVA table without observed values. I only managed to find SST with the first line of hint, but do not know how to proceed to find SSTreatment or SSError. Any hint/solution will be greatly appreciated.

Answer 1

First set up the error funcion

$$F(\beta_0, \beta_1) =\sum_{i=1}^{N}[y_i-\beta_0-\beta_1x_i]^2.$$

Differentiate with respect to $\beta_0$ and $\beta_1$ and set the derivatives equal to $0$ to obtain.

$$\sum_{i=1}^Ny_i-N\beta_0-\left[\sum_{i=1}^{N}x_i\right]\beta_1=0$$ $$\sum_{i=1}^Ny_ix_i-\left[\sum_{i=1}^{N}x_i\right]\beta_0-\left[\sum_{i=1}^{N}x^2_i\right]\beta_1=0.$$

The first equation can be rewritten as

$$N\bar{y}-N\beta_0-N\bar{x}\beta_1=0 \implies \beta_0 = \bar{y}-2\bar{x}=0.$$

From the second equaiton we can obtain

$$150-15\cdot 2\cdot 0 - \left[\sum_{i=1}^{15}x^2_i\right]\beta_1=0$$ $$\implies \sum_{i=1}^{15}x^2_i=37.5$$

We know that the correlation $r$ is given by

$$r= \dfrac{1/N\sum_{i=1}^{15}x_iy_i-\bar{x}\bar{y}}{s_x s_y}$$

We have to determine $s_x$ in order to calculate $r$: $$s_x = \dfrac{1}{15-1}\sum_{i=1}^{15}(x_i-\bar{x})^2$$ $$= \dfrac{1}{15-1}\left[\sum_{i=1}^{15}x^2_i-2\bar{x}\sum_{i=1}^{15}x_i+\sum_{i=1}^{15}\bar{x}^2\right]$$ $$= \dfrac{1}{15-1}\left[\sum_{i=1}^{15}x^2_i-2\cdot 15\cdot\bar{x}^2+15\bar{x}^2\right]$$ $$= \dfrac{1}{15-1}\left[\sum_{i=1}^{15}x^2_i-15\cdot\bar{x}^2\right].$$

Finally, we know that $\text{SSTotal}=\sum_{i=1}^{15}\left[y_i-\bar{y}\right]^2$ and

$$r^2 = \dfrac{\text{SSTreatment}}{\text{SSTotal}}=\dfrac{\text{SSTreatment}}{\sum_{i=1}^{15}\left[y_i-\bar{y}\right]^2}=\dfrac{\text{SSTreatment}}{(15-1)s^2_y}$$ $$\implies \text{SSTreatment} =(15-1)\,r^2\,s^2_y$$