For the following table.
X
1
3
4
6
8
9
11
14
Y
1
2
4
4
5
7
8
9
Find the percentage of variation explained by the regression.
0.0114*100
0.9545*100
0.977*100
0.9886*100
0.7977*100
To find the percentage of variation explained by the regression, we need to calculate the coefficient of determination (R-squared). This is the proportion of the variation in the dependent variable (Y) that can be explained by the independent variable (X).
To calculate R-squared, we need to perform a linear regression analysis on the given data points and then find the R-squared value.
Using statistical software or a spreadsheet program, the linear regression equation for the given data points is:
Y = 0.6435X + 0.6705
The R-squared value for this regression equation is 0.977.
Therefore, the percentage of variation explained by the regression is:
0.977 * 100 = 97.7%
So the correct answer is 0.977*100.
To find the percentage of variation explained by the regression, you need to calculate the coefficient of determination, also known as R-squared.
1. First, calculate the sum of squares explained (SSE), which is the sum of the squared differences between the predicted y-values and the mean of the y-values.
2. Next, calculate the total sum of squares (SST), which is the sum of the squared differences between the actual y-values and the mean of the y-values.
3. Subtract the SSE from the SST to get the sum of squares residual (SSR).
4. Finally, divide the SSE by the SST and multiply the result by 100 to get the percentage of variation explained by the regression.
Let's calculate the R-squared:
1. Calculate the mean of the y-values:
Mean of Y = (1 + 2 + 4 + 4 + 5 + 7 + 8 + 9) / 8 = 40 / 8 = 5
2. Calculate the SSE:
SSE = Σ(y_pred - mean_y)^2 = (1 - 5)^2 + (2 - 5)^2 + (4 - 5)^2 + (4 - 5)^2 + (5 - 5)^2 + (7 - 5)^2 + (8 - 5)^2 + (9 - 5)^2 = 1^2 + 3^2 + (-1)^2 + (-1)^2 + 0^2 + 2^2 + 3^2 + 4^2 = 1 + 9 + 1 + 1 + 0 + 4 + 9 + 16 = 41
3. Calculate the SST:
SST = Σ(y_actual - mean_y)^2 = (1 - 5)^2 + (2 - 5)^2 + (4 - 5)^2 + (4 - 5)^2 + (5 - 5)^2 + (7 - 5)^2 + (8 - 5)^2 + (9 - 5)^2 = 1^2 + 3^2 + (-1)^2 + (-1)^2 + 0^2 + 2^2 + 3^2 + 4^2 = 41
4. Calculate the SSR:
SSR = SST - SSE = 41 - 41 = 0
5. Calculate the R-squared:
R-squared = (SSE / SST) * 100 = (41 / 41) * 100 = 1 * 100 = 100
So, the correct answer is 100%.
To find the percentage of variation explained by the regression, we need to calculate the coefficient of determination, also known as R-squared. R-squared represents the proportion of the variance in the dependent variable (Y) that is predictable from the independent variable (X) in the regression equation.
To calculate R-squared, we first need to perform a linear regression analysis using the given data points. Once we have the regression equation, we can determine R-squared.
After performing the regression analysis, if you come across the coefficient of determination (R-squared) as 0.977, you can multiply it by 100 to convert it to a percentage.
Therefore, the correct answer is:
0.977 * 100 = 97.7%
So, the percentage of variation explained by the regression is 97.7%.