Q.4.134

Determine the linear correlation coefficient by using

a. Definition 4.8 on page 183

b. Formula 4.3 on page 185

Verified

Answer

(a) The linear correlation coefficient by using definition is $0.5$

(b) The linear correlation coefficient by using Formula is $0.5$

1Part(a) Step 1: Given Information

Given data is

We have to obtain the linear correlation coefficient by using definition

2Part(a) Step 2: Explanation

The linear correlation coefficient using definition is as follows

$r = \frac{\frac{1}{n - 1} \sum (x_{i} - \bar{x}) (y_{i} - \bar{y})}{σ_{x} σ_{y}}$

The standard deviations are as follows

$σ_{x} = \sqrt{\frac{1}{n - 1} \sum (x - \bar{x})^{2}} σ_{y} = \sqrt{\frac{1}{n - 1} \sum (y - \bar{y})^{2}}$

Table of values can be computed on the following table

The standard deviations can be calculated as follows

$σ_{x} = \sqrt{\frac{1}{n - 1} \sum (x - \bar{x})^{2}} = \sqrt{\frac{1}{3 - 1} (2)} = 1$

$σ_{y} = \sqrt{\frac{1}{n - 1} \sum (y - \bar{y})^{2}} = \sqrt{\frac{1}{3 - 1} (8)} = 2$

The correlation coefficient can be calculated as follows

$r = \frac{\frac{1}{n - 1} \sum (x_{i} - \bar{x}) (y_{i} - \bar{y})}{σ_{x} σ_{y}} = \frac{\frac{1}{3 - 1} (2)}{(1) (2)} = 0.5$

3Part(b) Step 1: Given Information

Given data is

We have to obtain the linear correlation coefficient by using formula

4Part(b) Step 2: Explanation

Table of values can be computed on the following table

The linear correlation coefficient using formula is as follows

r = \frac{\sum x y - \frac{\sum x \sum y}{n}}{\sqrt{(\sum x^{2} - \frac{\sum x^{2}}{n}) (\sum y^{2} - \frac{\sum y^{2}}{n})}} = \frac{47 - \frac{9 (15)}{3}}{\sqrt{(29 - (29 / 3)) (83 - (83 / 3))}} = 0.5