Regression analysis and Covariance

Regression analysis is a statistical tool that helps to investigate the relationship between a dependent variable and one or more independent variables. It is used to develop an equation that can be used to predict the value of the dependent variable based on the values of the independent variables.

On the other hand, covariance is a statistical measure that shows how much two variables change together. It measures the strength and direction of the linear relationship between two variables.

Regression equation

The regression equation of Y on X is given as 8X – 10Y = 66. This equation can be rearranged to solve for Y as follows:

8X – 10Y = 66
-10Y = -8X + 66
Y = (8/10)X – (66/10)
Y = 0.8X – 6.6

This equation shows that for every one-unit increase in X, Y is expected to increase by 0.8 units.

Standard deviation of X

The standard deviation of X is given as 3. This means that the values of X are spread out around the mean by an average of 3 units.

Covariance

The formula for covariance is given as:

Cov(X,Y) = Σ[(Xi – X)(Yi – Y)] / (n-1)

Where:
Xi = the ith value of X
X = the mean of X
Yi = the ith value of Y
Y = the mean of Y
n = the number of observations

Since the values of X and Y are not given, we cannot compute the covariance directly. However, we can use the regression equation to estimate the value of Y for each value of X.

Estimation of Covariance

Using the regression equation Y = 0.8X – 6.6, we can estimate the value of Y for each value of X.

For example, when X = 1, Y = 0.8(1) – 6.6 = -5.8
When X = 2, Y = 0.8(2) – 6.6 = -5.0
When X = 3, Y = 0.8(3) – 6.6 = -4.2

We can then use these values to compute the covariance as follows:

Cov(X,Y) = Σ[(Xi – X)(Yi – Y)] / (n-1)
Cov(X,Y) = [(1-2.0)(-5.8+5.4) + (2-2.0)(-5.0+5.4) + (3-2.0)(-4.2+5.4)] / (3-1)
Cov(X,Y) = [-0.4 + 0.8 + 1.2] / 2
Cov(X,Y) = 0.8

Therefore, the value of Cov(X,Y) is 0.8. This indicates a positive linear relationship between X and Y, meaning that as X increases, Y tends to increase as well.

YonX = 8x-10y+66=0

y = 8x/10-66/10

bxy = 8/10= 4/5

byx = cov(x,y)/SDx

4/5 = cov(x,y)/3

cov(x,y) = 12/5