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.
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.
To calculate the percentage of variation explained by the regression, you will need to perform a regression analysis and obtain the coefficient of determination (R-squared). R-squared is a statistical measure that represents the proportion of the dependent variable's variance that can be predicted by the independent variable(s). It indicates the strength of the relationship between the variables.
To calculate R-squared, you can follow these steps:
1. First, plot the given data points on a scatter plot with X values on the x-axis and Y values on the y-axis.
2. Use a regression analysis method (such as linear regression) to fit a regression line to the data.
3. Calculate the sum of squares total (SST), which represents the total variation in the Y values. It can be calculated by finding the sum of the squared differences between each Y value and the mean of the Y values.
4. Calculate the sum of squares regression (SSR), which represents the variation in the Y values that is explained by the regression line. It can be calculated by finding the sum of the squared differences between each predicted Y value (obtained from the regression line) and the mean of the Y values.
5. Calculate the coefficient of determination (R-squared) by dividing SSR by SST: R-squared = SSR / SST.
6. Finally, multiply R-squared by 100 to obtain the percentage of variation explained by the regression.
Please note that to perform these calculations, you will need to use statistical software or programming language that supports regression analysis, such as Python with libraries like NumPy, SciPy, or statsmodels.
To calculate the percentage of variation explained by the regression, we need to calculate the coefficient of determination (R-squared). R-squared measures the proportion of the total variation in the dependent variable (Y) that is explained by the independent variable (X).
First, we need to perform linear regression on the given data points. Using a statistical software or Excel, we can obtain the regression equation:
Y = 0.1818X + 0.7273
Next, we need to calculate the sum of squares total (SST) and the sum of squares residual (SSR). SST represents the total variation in Y, while SSR represents the unexplained variation in Y by the regression.
SST = Σ(Y - Ȳ)^2
= (2 - 4.5)^2 + (3 - 4.5)^2 + (4 - 4.5)^2 + (6 - 4.5)^2 + (5 - 4.5)^2 + (8 - 4.5)^2
= 2.5 + 2.5 + 0.25 + 2.25 + 0.25 + 10.25
= 18
SSR = Σ(Y - Ŷ)^2
= (2 - Ŷ)^2 + (3 - Ŷ)^2 + (4 - Ŷ)^2 + (6 - Ŷ)^2 + (5 - Ŷ)^2 + (8 - Ŷ)^2
= (2 - (0.1818X + 0.7273))^2 + (3 - (0.1818X + 0.7273))^2 + (4 - (0.1818X + 0.7273))^2 + (6 - (0.1818X + 0.7273))^2 + (5 - (0.1818X + 0.7273))^2 + (8 - (0.1818X + 0.7273))^2
= (2 - 0.1818X - 0.7273)^2 + (3 - 0.1818X - 0.7273)^2 + (4 - 0.1818X - 0.7273)^2 + (6 - 0.1818X - 0.7273)^2 + (5 - 0.1818X - 0.7273)^2 + (8 - 0.1818X - 0.7273)^2
= (2 - 0.1818X - 0.7273)^2 + (3 - 0.1818X - 0.7273)^2 + (4 - 0.1818X - 0.7273)^2 + (6 - 0.1818X - 0.7273)^2 + (5 - 0.1818X - 0.7273)^2 + (8 - 0.1818X - 0.7273)^2
Once SST and SSR are calculated, the coefficient of determination (R-squared) can be calculated as:
R-squared = 1 - (SSR / SST)
Finally, to obtain the percentage of variation explained by the regression, we multiply the R-squared value by 100.
Percentage of variation explained = R-squared * 100
Please note that the above calculations require substituting the X values into the regression equation to obtain the corresponding Ŷ values.