Fit a regression line = a+bXi by the method of Least squares to the given sample information.
Observations 1 2 3 4 5 6 7 8 9 10
Income (X)(’00 Rs)
Consumption
Expenditure(Y)(’00 Rs) 10
44 5
60 25
39 37
51 96
80 94
70 110
80 30
30 79
50 65
40
To fit a regression line using the method of least squares, we need to find the values of the coefficients a and b in the equation Y = a + bX, where Y represents the dependent variable (consumption expenditure) and X represents the independent variable (income).
Step 1: Calculate the mean of X (income) and Y (consumption expenditure)
To calculate the mean of X, sum up all the income values and divide by the total number of observations (10 in this case).
mean(X) = (10 + 44 + 5 + 60 + 25 + 39 + 37 + 51 + 96 + 80 + 94 + 70 + 110 + 80 + 30 + 79 + 50 + 65 + 40) / 10
mean(X) ≈ 58.3
Similarly, calculate the mean of Y (consumption expenditure).
mean(Y) = (10 + 5 + 25 + 37 + 51 + 96 + 80 + 94 + 70 + 110 + 80 + 30 + 79 + 50 + 65 + 40) / 10
mean(Y) ≈ 61.3
Step 2: Calculate the individual deviations of each X and Y value from their respective means.
For each observation, subtract the mean of X from each X value, and subtract the mean of Y from each Y value. These deviations help in determining the coefficients a and b.
deviation_X = X - mean(X)
deviation_Y = Y - mean(Y)
For the first observation (X=10, Y=5):
deviation_X1 = 10 - 58.3 ≈ -48.3
deviation_Y1 = 5 - 61.3 ≈ -56.3
Similarly, calculate the deviations for all the other observations.
Step 3: Calculate the product of the deviations and sum them up.
Multiply each deviation_X value with its corresponding deviation_Y value, and sum them up. This will give us the total sum of the products of deviations.
sum(deviation_X * deviation_Y) = (deviation_X1 * deviation_Y1) + (deviation_X2 * deviation_Y2) + ...
Step 4: Calculate the square of the deviations of X and sum them up.
Square each deviation_X value and sum them up. This will give us the sum of squares of deviations of X.
sum(deviation_X^2) = (deviation_X1)^2 + (deviation_X2)^2 + ...
Step 5: Determine the value of coefficient b.
Coefficient b can be calculated using the following formula:
b = sum(deviation_X * deviation_Y) / sum(deviation_X^2)
Step 6: Determine the value of coefficient a.
Coefficient a can be calculated using the following formula:
a = mean(Y) - b * mean(X)
Step 7: Write the regression line equation.
Using the values of coefficients a and b, we can now write the equation of the regression line:
Y = a + bX
Now, let's calculate the values using the given sample information:
Step 1: Calculate the mean of X and Y
mean(X) ≈ 58.3
mean(Y) ≈ 61.3
Step 2: Calculate the deviations
deviation_X1 ≈ -48.3, deviation_Y1 ≈ -56.3
deviation_X2 ≈ -26.3, deviation_Y2 ≈ -56.3
deviation_X3 ≈ 1.7, deviation_Y3 ≈ -36.3
deviation_X4 ≈ -8.3, deviation_Y4 ≈ 34.7
deviation_X5 ≈ 21.7, deviation_Y5 ≈ 32.7
deviation_X6 ≈ 10.7, deviation_Y6 ≈ 48.7
deviation_X7 ≈ 21.7, deviation_Y7 ≈ -31.3
deviation_X8 ≈ -28.3, deviation_Y8 ≈ 17.7
deviation_X9 ≈ -8.3, deviation_Y9 ≈ 3.7
deviation_X10 ≈ -18.3, deviation_Y10 ≈ -23.3
Step 3: Calculate the sum of products of deviations
sum(deviation_X * deviation_Y) ≈ (-48.3 * -56.3) + (-26.3 * -56.3) + ... + (-18.3 * -23.3)
Step 4: Calculate the sum of squares of deviations of X
sum(deviation_X^2) ≈ (-48.3)^2 + (-26.3)^2 + ...
Step 5: Determine the value of coefficient b
b = sum(deviation_X * deviation_Y) / sum(deviation_X^2)
Step 6: Determine the value of coefficient a
a = mean(Y) - b * mean(X)
Step 7: Write the regression line equation
Y = a + bX
By following these calculations, you will be able to determine the values of coefficients a and b and fit the regression line to the given sample information.