91Ó°ÊÓ

To find the linear correlation coefficient by the definition.

The linear correlation coefficient using the definition is

$r=\frac{\frac{1}{n-1}∑\left(x_i-\stackrel{¯}{x}\right)\left(y_i-\stackrel{¯}{y}\right)}{{\mathrm{σ}}_x{\mathrm{σ}}_y}$

The standard deviations are

$$\begin{gathered} {\mathrm{σ}}_x=\sqrt{\frac{1}{n-1}∑(x-\stackrel{¯}{x}{)}^2}, \\ {\mathrm{σ}}_y=\sqrt{\frac{1}{n-1}∑(y-\stackrel{¯}{y}{)}^2} \end{gathered}$$

Calculation:

Make a table of values

Find the standard deviations

$$\begin{gathered} {\mathrm{σ}}_x=\sqrt{\frac{1}{n-1}∑(x-\stackrel{¯}{x}{)}^2} \\ =\sqrt{\frac{1}{3-1}\left(2\right)} \\ =1 \\ {\mathrm{σ}}_y=\sqrt{\frac{1}{n-1}∑(y-\stackrel{¯}{y}{)}^2} \\ =\sqrt{\frac{1}{3-1}\left(14\right)} \\ =\sqrt{7} \end{gathered}$$

Find the correlation coefficient

$r=\frac{\frac{1}{n-1}∑\left(x_i-\stackrel{¯}{x}\right)\left(y_i-\stackrel{¯}{y}\right)}{{\mathrm{σ}}_x{\mathrm{σ}}_y}$

$=\frac{\frac{1}{3-1}(-4)}{\left(1\right)\left(\sqrt{7}\right)}$

Hence,

The linear coefficient by the definition$=-0.759$

To find the linear correlation coefficient by the formula.

The linear correlation coefficient using the formula

$r=\frac{∑xy-\frac{{∑}^x{∑}^y}{n}}{\sqrt{\left(∑x^2-\frac{{∑}^2}{n}\right)\left(∑y^2-\frac{{∑}^2}{n}\right)}}$

Calculation:

Make a table of values

Find the correlation coefficient

localid="1650908959084" role="math" $$\begin{gathered} r=\frac{∑xy-\frac{∑x∑y}{n}}{\sqrt{\left(∑x^2-\frac{{\left(∑x\right)}^2}{n}\right)\left(∑y^2-\frac{{\left(∑y\right)}^2}{n}\right)}} \\ =\frac{-22-\frac{6(-9)}{3}}{\sqrt{(14-(36/3\left)\right)(41-(81/3\left)\right)}} \end{gathered}$$

Hence,

The linear coefficient by the formula$=-0.756$