Calculation of Correlation Coefficient using Regression Lines

To calculate the correlation coefficient using regression lines, we need to follow the below steps:

1. Find the slopes of both regression lines.
2. Use the formula r = √(b1 * b2) to calculate the correlation coefficient where b1 and b2 are the slopes of the two regression lines.

Finding the Slopes of Regression Lines

The given regression lines are:

10x - 20y + 132 = 0
80x - 30y - 428 = 0

To find the slopes of the regression lines, we need to write them in the form y = mx + c where m is the slope.

The first regression line can be written as:

10x - 20y + 132 = 0
-20y = -10x + 132
y = (1/2)x - (132/20)
y = (1/2)x - 6.6

Therefore, the slope of the first regression line is 1/2.

Similarly, the second regression line can be written as:

80x - 30y - 428 = 0
-30y = -80x + 428
y = (8/3)x - (428/30)
y = (8/3)x - 14.27

Therefore, the slope of the second regression line is 8/3.

Calculation of Correlation Coefficient

Now, we can use the formula r = √(b1 * b2) to calculate the correlation coefficient where b1 and b2 are the slopes of the two regression lines.

r = √[(1/2) * (8/3)]
r = √(4/3)
r = 0.816

Therefore, the correlation coefficient is 0.816.

Explanation of Correlation Coefficient

The correlation coefficient is a measure of the strength and direction of the linear relationship between two variables. It ranges between -1 and +1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and +1 indicates a perfect positive correlation.

In this case, the correlation coefficient is 0.816, which indicates a strong positive correlation between the two variables. This means that as one variable increases, the other variable also tends to increase. The value of the correlation coefficient suggests a strong linear relationship between the two variables.