Question

Calculate SSE andfor each of the following cases:

a. n = 20,$\mathit{S}{\mathit{S}}_{\mathbf{y}\mathbf{y}}\mathbf{=}\mathbf{95}$,$\mathit{S}{\mathit{S}}_{\mathbf{x}\mathbf{y}}\mathbf{=}\mathbf{50}$,$\widehat{{\mathbf{\beta }}_{\mathbf{1}}}\mathbf{=}\mathbf{.}\mathbf{75}$

b. n = 40,$\mathbf{\sum }{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{860}$ , $\mathbf{\sum }\mathit{y}\mathbf{=}\mathbf{50}$, $\mathit{S}{\mathit{S}}_{\mathbf{x}\mathbf{y}}\mathbf{=}\mathbf{2700}$, $\widehat{{\mathbf{\beta }}_{\mathbf{1}}}\mathbf{=}\mathbf{.}\mathbf{2}$

c. n = 10, $\mathbf{\sum }{\left(y_i-\overline{y}\right)}^{\mathbf{2}}\mathbf{=}\mathbf{58}$,$\mathit{S}{\mathit{S}}_{\mathbf{x}\mathbf{y}}\mathbf{=}\mathbf{91}$,$\mathit{S}{\mathit{S}}_{\mathbf{x}\mathbf{x}}\mathbf{=}\mathbf{170}$

Short answer

1. SSE = 57.5, ${\mathit{s}}^{\mathbf{2}}$= 3.195
2. SSE = 257.5,${\mathit{s}}^{\mathbf{2}}$ = 6.78
3. SSE = 12.5, ${\mathit{s}}^{\mathbf{2}}$= 1.56

Step-by-step solution

Introduction

The overall deviation of the response values from the response values' fit is calculated using this statistic. The summed square of residuals is abbreviated as SSE.

Find SSE and s2for n = 20, role="math" localid="1668605135813" SSyy=95,role="math" localid="1668605146951" SSxy=50  ,  role="math" localid="1668605158273" β1^=.75

\(\begin{aligned}{c}SSE &= S{S_{yy}} - \widehat {{\beta _1}}S{S_{xy}}\\ &= 95 - (.75)(50)\\ &= 95 - 37.5\\ &= 57.5\end{aligned}\)

\(\begin{aligned}{c} {s^2} &= \frac{{SSE}}{{n - 2}}\\ &= \frac{{57.5}}{{20 - 2}}\\ &= \frac{{57.5}}{{18}}\\ &= 3.195\end{aligned}\)

Therefore,SSE is 57.5 and \( {s^2}\) 3.195.

Find SSE and s2for n = 40,∑y2=860 , ∑y=50 , SSxy=2700 ,  β1^=.2

\(\begin{aligned}{c}S{S_{yy}} &= \sum {y^2} - \frac{{{{\left( {\sum y} \right)}^2}}}{n}\\ &= 860 - \frac{{{{\left( {50} \right)}^2}}}{{40}}\\ &= 860 - \frac{{2500}}{{40}}\\ &= 860 - 62.5\end{aligned}\)

\( = 797.5\)

\(\begin{aligned}{c}SSE &= S{S_{yy}} - \widehat {{\beta _1}}S{S_{xy}}\\ &= 797.5 - (.2)(2700)\\ &= 797.5 - 540\\ &= 257.5\end{aligned}\)

\(\begin{aligned}{c} {s^2} &= \frac{{SSE}}{{n - 2}}\\ &= \frac{{257.5}}{{40 - 2}}\\ &= \frac{{257.5}}{{38}}\\ &= 6.78\end{aligned}\)

Therefore, SSE is 257.5 and ${\mathit{s}}^{\mathbf{2}}$6.78.

Find SSE and s2for n = 10,  ∑(yi-y¯)2=58, SSxy=91, SSxx=170

\(S{S_{yy}} = \sum {\left( {{y_i} - \overline y } \right)^2} = 58\)

\(\begin{aligned}{c}\widehat {{\beta _1}} &= \frac{{S{S_{xy}}}}{{S{S_{xx}}}}\\ &= \frac{{91}}{{170}}\\ &= 0.5\end{aligned}\)

\(\begin{aligned}{c}SSE &= S{S_{yy}} - \widehat {{\beta _1}}S{S_{xy}}\\ &= 58 - (.5)(91)\\ &= 58 - 45.5\\ &= 12.5\end{aligned}\)

\(\begin{aligned}{c} {s^2} &= \frac{{SSE}}{{n - 2}}\\ &= \frac{{12.5}}{{10 - 2}}\\ &= \frac{{12.5}}{8}\\ &= 1.56\end{aligned}\)

Therefore, SSE is 12.5 and ${\mathit{s}}^{\mathbf{2}}$1.56.