Question

Determine the linear correlation coefficient by formula and definition.

Short answer

a) The linear coefficient by the definition $=-0.721$

b) The linear coefficient by the formula$=-0.721$

Step-by-step solution

Part (a) Step 1: Given Information

To find the linear correlation coefficient by the definition.

Part (a) Step 2: Explanation

The linear correlation coefficient using the definition is

$r=\frac{\frac{1}{n-1}\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{{\sigma }_x{\sigma }_y}$

The standard deviations are

$$\begin{gathered} {\sigma }_x=\sqrt{\frac{1}{n-1}\sum (x-\overline{x}{)}^2}, \\ {\sigma }_y=\sqrt{\frac{1}{n-1}\sum (y-\overline{y}{)}^2} \end{gathered}$$

Calculation:

Make a table of values

Find the standard deviations

$$\begin{gathered} {\sigma }_x=\sqrt{\frac{1}{n-1}\sum (x-\overline{x}{)}^2} \\ =\sqrt{\frac{1}{4-1}\left(6\right)} \\ =1.414 \end{gathered}$$

$$\begin{gathered} {\sigma }_y=\sqrt{\frac{1}{n-1}\sum (y-\overline{y}{)}^2} \\ =\sqrt{\frac{1}{4-1}\left(104\right)} \\ =5.88 \end{gathered}$$

Find the correlation coefficient

$$\begin{gathered} r=\frac{\frac{1}{n-1}\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)}{{\sigma }_{\overline{x}}{\sigma }_y} \\ =\frac{\frac{1}{4-1}(-18)}{(1.414)(5.88)} \end{gathered}$$

Hence,

The linear coefficient by the definition$=-0.721$

Part (b) Step 1: Given Information

To find the linear correlation coefficient by the formula.

Part (b) tep 2: Explanation

The linear correlation coefficient using the formula

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

Calculation:

Make a table of values

Find the correlation coefficient

$$\begin{gathered} r=\frac{\sum xy-\frac{\sum x{\sum }^n}{n}}{\sqrt{\left(\sum x^2-\frac{{\left(\sum x\right)}^2}{n}\right)\left(\sum y^2-\frac{{\left(\sum y\right)}^2}{n}\right)}} \\ =\frac{42-\frac{240}{4}}{\sqrt{(42-(144/4\left)\right)(204-(20\sqrt{4}\left)\right)}} \end{gathered}$$

Hence,

The linear coefficient by the formula$=-0.721$

Hence, both the results are the same.