Regression Coefficient and Line Regression

Regression analysis is a statistical technique used to determine the relationship between two or more variables. It helps in understanding how the value of a dependent variable changes when one or more independent variables are varied.

In this case, we have the following data:
X: 1, 2, 3, 4, 5, 6, 7
Y: 9, 8, 10, 12, 11, 13, 14

To calculate the regression coefficient and obtain the line regression, we will follow these steps:

Step 1: Calculate the mean of X and Y
To calculate the mean of X, we sum all the values of X and divide by the total number of values:
Mean of X = (1 + 2 + 3 + 4 + 5 + 6 + 7) / 7 = 4

Similarly, to calculate the mean of Y:
Mean of Y = (9 + 8 + 10 + 12 + 11 + 13 + 14) / 7 = 11

Step 2: Calculate the deviations from the mean
Next, we calculate the deviations of each X value from the mean of X, and each Y value from the mean of Y:

Deviation of X = (X - Mean of X)
Deviation of Y = (Y - Mean of Y)

For example, the deviation of X for the first value (X=1) is:
Deviation of X = 1 - 4 = -3

Step 3: Calculate the product of deviations
We multiply the deviations of X and Y for each data point:

Product of deviations = (Deviation of X) * (Deviation of Y)

For example, for the first data point, the product of deviations is:
Product of deviations = (-3) * (9 - 11) = 6

Step 4: Calculate the sum of products of deviations
We sum all the products of deviations for all data points:

Sum of products of deviations = Σ(Product of deviations)

Step 5: Calculate the sum of squared deviations of X
We square the deviations of X for each data point and sum them:

Sum of squared deviations of X = Σ(Deviation of X)^2

Step 6: Calculate the regression coefficient (b)
Finally, we calculate the regression coefficient using the following formula:

b = Sum of products of deviations / Sum of squared deviations of X

Step 7: Obtain the line regression equation
Once we have the regression coefficient (b), we can obtain the line regression equation:

Y = a + bX

Where 'a' is the intercept, which can be calculated using the following formula:

a = Mean of Y - b * Mean of X

Step 8: Interpret the line regression equation
The line regression equation provides the relationship between the independent variable (X) and the dependent variable (Y). The coefficient 'b' represents the change in Y for every unit change in X. The intercept 'a' represents the value of Y when X is equal to zero.

In conclusion, by following the above steps, we can calculate the regression coefficient and obtain the line regression equation to understand the relationship between