Question

In order to find the correlation coefficient between two variables X and Y
from 20 pairs of observations, the following calculations were made:
Σx =15, Σy = − 6, Σxy = 50, Σx^2 = 61 and Σy^2 = 90 .
Calculate the correlation coefficient, and the slope of the regression line of Y on X.

Answer 1

Answer on Question #46486 - Math - Statistics and Probability

In order to find the correlation coefficient between two variables X and Y from 20 pairs of observations, the following calculations were made:

$$\Sigma x = 15, \Sigma y = -6, \Sigma xy = 50, \Sigma x^2 = 61 \text{ and } \Sigma y^2 = 90.$$

Calculate the correlation coefficient, and the slope of the regression line of Y on X.

Solution

Correlation coefficient is calculated as follows:

$$r_{xy} = \frac{\sum x_i y_i - n \hat{x} \hat{y}}{(n-1) s_x s_y} = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{\sqrt{n \sum x_i^2 - (\sum x_i)^2} \sqrt{n \sum y_i^2 - (\sum y_i)^2}}$$ $$r_{xy} = \frac{20 \cdot 50 - 15 \cdot (-6)}{\sqrt{20 \cdot 61 - 15^2} \cdot \sqrt{20 \cdot 90 - 6^2}} = 0.823$$

The slope of the regression line of Y on X is calculated as follows:

$$a = r \frac{s_x}{s_y} = \frac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - (\sum x_i)^2} = \frac{20 \cdot 50 - 15 \cdot -6}{20 \cdot 61 - 15^2} = 1.095$$

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