Relationship between regression coefficient and correlation coefficient
Answers
Answer:
As we have seen in the previous pages, the variables x and y can be linearly related. The correlation coefficient is a measure of the degree of relationship present between the linearly related variables.
Example:
Values of x Values of y
1 1
2 2
3 3
4 4
5 5
6 6
From the graph, we observe that the slope of the line is 1.00. There is a perfect positive relationship between x values and y values and we can deduce that the correlation coefficient is 1.00.
What is correlation coefficient ?
The correlation coefficient is defined as the covariance of x and y divided by the product of the standard deviations of x and y and which can be represented in the following form :
We have seen from the cost modeller example and the SAT scores example that the x values and y values will not vary in the same fashion. Both the x and y values are subject to variation. So, under these circumstances, there exists not one regression line, but, two regression lines. We have situations in which y is considered to be a dependent variable and x is taken as an independent variable or other times x is considered to be the dependent variable and y is taken as independent.
What is regression coefficient ?
The Regression coefficient is defined as the covariance of x and y divided by the variance of the independent variable, x or y.
In other words, the regression coefficient of y on x is defined as the covariance of x and y divided by the variance of the independent variable, x which can be represented in the following form :
Similarily, the regression coefficient of x on y can be written as follows :
Suppose we multiply the above two equations, i.e.,if we multiply the two regression coefficients that are given above, we obtain:
We observe that both the regression coefficents must have the same sign and that the sign is determined by r. The value r2 is referred to as the coefficient of determination. The value 1 - r2 is referred to as the coefficient of nondetermination.
Step-by-step explanation:
Answer:
Step-by-step explanation:
