Regression Coefficient

The regression coefficient measures the change in the dependent variable (Y) for a unit change in the independent variable (X). In this case, the regression coefficient between X and Y is given as -2/3.

Y on X

The regression coefficient, Y on X, indicates the change in the dependent variable (Y) for a unit change in the independent variable (X). In this case, the coefficient Y on X is given as -1/6.

Coefficient of Correlation

The coefficient of correlation (r) measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1. A positive value indicates a positive linear relationship, while a negative value indicates a negative linear relationship.

Calculation of Coefficient of Correlation

To calculate the coefficient of correlation, we can use the relationship between the regression coefficient and the coefficient of correlation:

r = bYX * (σX / σY)

Where:
- bYX is the regression coefficient of Y on X (-1/6)
- σX is the standard deviation of X
- σY is the standard deviation of Y

Calculating Standard Deviations

To calculate the standard deviations, we need additional information. Let's assume we have the data for X and Y and calculate the standard deviations.

Example:
Assume we have the following data:

X: 1, 2, 3, 4, 5
Y: 3, 5, 7, 9, 11

Calculating Standard Deviation of X (σX)
1. Calculate the mean of X:
Mean(X) = (1 + 2 + 3 + 4 + 5) / 5 = 3

2. Calculate the deviations of each value of X from the mean:
Deviations(X) = X - Mean(X)
= (1 - 3, 2 - 3, 3 - 3, 4 - 3, 5 - 3)
= (-2, -1, 0, 1, 2)

3. Square each deviation:
Squared Deviations(X) = (-2)^2, (-1)^2, 0^2, 1^2, 2^2
= (4, 1, 0, 1, 4)

4. Calculate the mean of the squared deviations:
Mean(Squared Deviations(X)) = (4 + 1 + 0 + 1 + 4) / 5
= 2

5. Take the square root of the mean of squared deviations:
σX = sqrt(Mean(Squared Deviations(X)))
= sqrt(2)

Calculating Standard Deviation of Y (σY)
Using the same steps as above, we can calculate the standard deviation of Y:

1. Calculate the mean of Y:
Mean(Y) = (3 + 5 + 7 + 9 + 11) / 5
= 7

2. Calculate the deviations of each value of Y from the mean:
Deviations(Y) = Y - Mean(Y)
= (3 - 7, 5 - 7, 7

Minus 1 by 3