Final answer:
To determine the fraction of the variation in y explained by x, we need to calculate the coefficient of determination (r²). This can be done using the correlation coefficient and the regression equation. The coefficient of determination represents the proportion of the total variation in y that can be explained by the regression line.
Step-by-step explanation:
To determine what fraction of the variation in y can be explained by the variation in x, we need to calculate the coefficient of determination (r²).
The coefficient of determination (r²) represents the proportion of the total variation in y that can be explained by the regression line. It ranges from 0 to 1, where 0 indicates no relationship and 1 indicates a perfect fit.
In this case, the regression equation is y = -5.7 + 1.74x. Since the coefficient of determination is equal to the square of the correlation coefficient (r), we can square the correlation coefficient to find r².
Given that the standard deviation for x is 1.56 and the standard deviation for y is 5.46, we can use these values to calculate the correlation coefficient, which is then squared to find r².
Using a statistics software package, the correlation coefficient can be calculated by dividing the covariance of x and y by the product of their standard deviations.
Once r² is determined, it represents the fraction of variation in y that can be explained by the variation in x. For example, if r² is 0.5, then 50% of the variation in y can be explained by the variation in x.