To calculate the percentage of variation explained by the regression, we need to find the coefficient of determination (r-squared value). This can be calculated using the formula:
r^2 = (SSR / SST)
where SSR is the sum of squares due to regression and SST is the total sum of squares.
First, we need to find the regression line that fits the given data. Assuming a linear regression model, we can use the least squares method to find the equation of the line:
Y = a + bX
where a is the y-intercept and b is the slope.
Using the given data points (X and Y), we can calculate the following:
X̄ = (3 + 5 + 6 + 8 + 9 + 11) / 6 = 7
Ȳ = (2 + 3 + 4 + 6 + 5 + 8) / 6 = 4.67
Using the formulas for the slope and the y-intercept:
b = Σ((X - X̄)(Y - Ȳ)) / Σ((X - X̄)^2)
a = Ȳ - bX̄
Substituting the values:
b = ((3 - 7)(2 - 4.67) + (5 - 7)(3 - 4.67) + (6 - 7)(4 - 4.67) + (8 - 7)(6 - 4.67) + (9 - 7)(5 - 4.67) + (11 - 7)(8 - 4.67)) / ((3 - 7)^2 + (5 - 7)^2 + (6 - 7)^2 + (8 - 7)^2 + (9 - 7)^2 + (11 - 7)^2) = 0.836
a = 4.67 - 0.836(7) ≈ -0.67
Therefore, the equation of the regression line is:
Y = -0.67 + 0.836X
Next, we can calculate SSR:
SSR = Σ((Y - Ŷ)^2)
where Y is the actual Y value and Ŷ is the predicted Y value using the regression line.
Using the given data points and the equation of the regression line:
Ŷ = -0.67 + 0.836X
We get the following table:
X Y Ŷ (Y - Ŷ) (Y - Ŷ)^2
3 2 0.338 1.662 2.755
5 3 1.672 1.328 1.761
6 4 2.508 1.492 2.228
8 6 4.180 1.820 3.317
9 5 5.016 -0.016 0.000
11 8 6.688 1.312 1.722
SSR = Σ((Y - Ŷ)^2) = 11.783
Finally, we can calculate SST:
SST = Σ((Y - Ȳ)^2)
where Y is the actual Y value.
Using the given data points and the mean Y value (Ȳ = 4.67):
We get the following table:
X Y (Y - Ȳ) (Y - Ȳ)^2
3 2 -2.67 7.128
5 3 -1.67 2.788
6 4 -0.67 0.448
8 6 1.33 1.773
9 5 0.33 0.108
11 8 3.33 11.063
SST = Σ((Y - Ȳ)^2) = 23.309
Now, we can calculate the coefficient of determination (r-squared value):
r^2 = (SSR / SST) = (11.783 / 23.309) ≈ 0.5061
The percentage of variation explained by the regression is equal to the r-squared value multiplied by 100:
Percentage of variation explained = 0.5061 * 100 ≈ 50.61%
Therefore, the regression model explains approximately 50.61% of the variation in the data.
For the following table
X
3
5
6
8
9
11
Y
2
3
4
6
5
8
Calculate the persentage of variation explained by the regression.
1 answer