Question

Given the following data set, let x be the explanatory variable and y be the response variable.

x 1 1 7 8 6 5 8
y 10 9 3 2 6 7 3
(a) If a least squares line was fitted to this data, what percentage of the variation in the y would be explained by the regression line? (Enter your answer as a percent.)

Answer 1

To calculate the percentage of variation in the response variable (y) explained by the regression line, we need to find the coefficient of determination (r²). The coefficient of determination represents the proportion of the variation in y that can be explained by the variation in the explanatory variable (x). To do this, we need to calculate the correlation coefficient (r), square it to get r², and convert it to a percentage.

Step-by-step explanation:

In order to calculate the percentage of variation in the response variable (y) that is explained by the regression line, we need to find the coefficient of determination (r²). The coefficient of determination is the square of the correlation coefficient (r) and represents the proportion of the variation in y that can be explained by the variation in the explanatory variable (x).

• First, we need to calculate the correlation coefficient (r) by dividing the covariance of x and y by the product of their standard deviations.
• Next, we square the correlation coefficient (r) to get the coefficient of determination (r²).
• Finally, we convert the coefficient of determination (r²) to a percentage by multiplying it by 100.

Let's calculate the percentage of variation in y explained by the regression line:

r = covariance(x, y) / (standard deviation of x * standard deviation of y)

r = (-6.3) / (2.29 * 2.36) ≈ -0.570

r² = (-0.570)² ≈ 0.325

Percentage of variation explained = r² * 100 ≈ 32.5%