Question

Do the accompanying data provide sufficient evidence that a straight line is useful for characterizing the relationship between x and y?

X	4	2	4	3	2	4
Y	1	6	5	3	2	4

Short answer

The straight line is useful for defining the relationship between x and y at a =.05.

Step-by-step solution

Introduction

A straight line is the outcome of a simple linear regression study of two or more independent variables. A regression with numerous relevant variables might result in a curved line in rare instances.

Find t

$$\begin{gathered} \mathit{S}{\mathit{S}}_{\mathbf{x}\mathbf{y}}\mathbf{=}\mathbf{\sum }{\mathit{x}}_{\mathbf{i}}{\mathit{y}}_{\mathbf{i}}\mathbf{-}\frac{\mathbf{\sum }{\mathbf{x}}_{\mathbf{i}}\mathbf{\sum }{\mathbf{y}}_{\mathbf{i}}}{\mathbf{n}} \\ \mathbf{=}\mathbf{65}\mathbf{-}\frac{\mathbf{19}\mathbf{x}\mathbf{21}}{\mathbf{6}} \\ \mathbf{=}\mathbf{65}\mathbf{-}\frac{\mathbf{399}}{\mathbf{6}} \\ \mathbf{=}\frac{\mathbf{390}\mathbf{-}\mathbf{399}}{\mathbf{6}} \end{gathered}$$

$$\begin{gathered} \mathbf{=}\mathbf{-}\frac{\mathbf{9}}{\mathbf{6}} \\ \mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{5} \end{gathered}$$

$$\begin{gathered} \mathit{S}{\mathit{S}}_{\mathbf{x}\mathbf{x}}\mathbf{=}\mathbf{\sum }{{\mathbf{x}}_{\mathbf{i}}}^{\mathbf{2}}\mathbf{-}\frac{{\left(\sum x_i\right)}^{\mathbf{2}}}{\mathbf{n}} \\ \mathbf{=}\mathbf{65}\mathbf{-}\frac{{\left(19\right)}^{\mathbf{2}}}{\mathbf{6}} \\ \mathbf{=}\mathbf{65}\mathbf{-}\frac{\mathbf{361}}{\mathbf{6}} \\ \mathbf{=}\frac{\mathbf{390}\mathbf{-}\mathbf{361}}{\mathbf{6}} \end{gathered}$$

$$\begin{gathered} \mathbf{=}\frac{\mathbf{29}}{\mathbf{6}} \\ \mathbf{=}\mathbf{4}\mathbf{.}\mathbf{84} \end{gathered}$$

$$\begin{gathered} \widehat{{\mathbf{\beta }}_{\mathbf{1}}}\mathbf{=}\frac{\mathbf{S}{\mathbf{S}}_{\mathbf{x}\mathbf{y}}}{\mathbf{S}{\mathbf{S}}_{\mathbf{x}\mathbf{x}}} \\ \mathbf{=}\frac{\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{5}}{\mathbf{4}\mathbf{.}\mathbf{84}} \\ \mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{31} \end{gathered}$$

$$\begin{gathered} \mathit{S}{\mathit{S}}_{\mathbf{y}\mathbf{y}}\mathbf{=}\mathbf{\sum }{\mathit{y}}^{\mathbf{2}}\mathbf{-}\frac{{\left(\sum y\right)}^{\mathbf{2}}}{\mathbf{n}} \\ \mathbf{=}\mathbf{91}\mathbf{-}\frac{{\left(19\right)}^{\mathbf{2}}}{\mathbf{6}} \\ \mathbf{=}\mathbf{91}\mathbf{-}\frac{\mathbf{361}}{\mathbf{6}} \\ \mathbf{=}\frac{\mathbf{546}\mathbf{-}\mathbf{361}}{\mathbf{6}} \end{gathered}$$

$$\begin{gathered} \mathbf{=}\frac{\mathbf{185}}{\mathbf{6}} \\ \mathbf{=}\mathbf{30}\mathbf{.}\mathbf{84} \end{gathered}$$

$$\begin{gathered} \mathit{S}\mathit{S}\mathit{E}\mathbf{=}\mathit{S}{\mathit{S}}_{\mathbf{y}\mathbf{y}}\mathbf{-}\widehat{{\mathbf{\beta }}_{\mathbf{1}}}\mathit{S}{\mathit{S}}_{\mathbf{x}\mathbf{y}} \\ \mathbf{=}\mathbf{30}\mathbf{.}\mathbf{84}\mathbf{-}\mathbf{(}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{31}\mathbf{)}\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{)} \\ \mathbf{=}\mathbf{30}\mathbf{.}\mathbf{84}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{465} \\ \mathbf{=}\mathbf{30}\mathbf{.}\mathbf{375} \end{gathered}$$

$$\begin{gathered} {\mathit{s}}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{S}\mathbf{S}\mathbf{E}}{\mathbf{n}\mathbf{-}\mathbf{2}} \\ \mathbf{=}\frac{\mathbf{30}\mathbf{.}\mathbf{375}}{\mathbf{6}\mathbf{-}\mathbf{2}} \\ \mathbf{=}\frac{\mathbf{30}\mathbf{.}\mathbf{375}}{\mathbf{4}} \\ \mathbf{=}\mathbf{7}\mathbf{.}\mathbf{59375} \end{gathered}$$

$$\begin{gathered} \mathit{s}\mathbf{=}\sqrt{\frac{\mathbf{S}\mathbf{S}\mathbf{E}}{\mathbf{n}\mathbf{-}\mathbf{2}}} \\ \mathbf{=}\sqrt{\frac{\mathbf{30}\mathbf{.}\mathbf{375}}{\mathbf{6}\mathbf{-}\mathbf{2}}} \\ \mathbf{=}\sqrt{\frac{\mathbf{30}\mathbf{.}\mathbf{375}}{\mathbf{4}}} \\ \mathbf{=}\sqrt{\mathbf{7}\mathbf{.}\mathbf{59375}} \end{gathered}$$

$\mathbf{=}\mathbf{2}\mathbf{.}\mathbf{7557}$

$$\begin{gathered} \mathit{t}\mathbf{=}\frac{\widehat{{\mathbf{\beta }}_{\mathbf{1}}}}{\mathbf{s}\mathbf{/}\sqrt{\mathbf{S}{\mathbf{S}}_{\mathbf{x}\mathbf{x}}}} \\ \mathbf{=}\frac{\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{31}}{\mathbf{2}\mathbf{.}\mathbf{7557}\mathbf{/}\sqrt{\mathbf{4}\mathbf{.}\mathbf{84}}} \\ \mathbf{=}\mathbf{-}\frac{\mathbf{0}\mathbf{.}\mathbf{31}}{\mathbf{2}\mathbf{.}\mathbf{7557}\mathbf{/}\mathbf{2}\mathbf{.}\mathbf{2}} \\ \mathbf{=}\mathbf{-}\frac{\mathbf{0}\mathbf{.}\mathbf{31}}{\mathbf{1}\mathbf{.}\mathbf{2525}} \end{gathered}$$

$\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{3883}$

The rejection region is defined as t > 0.3883 or t -0.3883.

There is insufficient evidence to suggest that a straight line is suitable for defining the relationship between x and y at a =.05 since the observed value of the test statistic does not lie in the rejection region.