Question 7

The variable x explains 86% of the variation in the variable y.

The coefficient of determination, denoted as r², indicates that 86% of the variability in the dependent variable (y) can be explained by the independent variable (x). This strong correlation suggests a significant relationship between the two variables.

A) The ratio of the variable y to the variable x is approximately 0.86.

This option incorrectly interprets the coefficient of determination. r² does not represent a ratio of the two variables, but rather a measure of how well the independent variable explains the variability in the dependent variable.

B) The variable x explains 86% of the variation in the variable y.

This statement accurately reflects the meaning of the coefficient of determination. An r² value of 0.86 indicates that 86% of the variance in y can be accounted for by changes in x, making this the correct interpretation.

C) The total variation in one variable is 86% of the variation in the other variable.

This option misinterprets the relationship described by r². The coefficient of determination does not imply that the total variation in one variable is a percentage of the variation in the other variable; rather, it indicates the proportion of variance explained.

D) The linear regression equation has a slope of 0.86 or -0.86.

This statement is incorrect as it conflates the coefficient of determination with the slope of the regression line. The value of r² does not provide any information about the slope, which is determined separately in the regression analysis.

Conclusion

The correct interpretation of the coefficient of determination is that variable x explains 86% of the variation in variable y, as stated in option B. The other options fail to accurately describe the meaning of r², either by misrepresenting its function or confusing it with other statistical measures. This highlights the importance of understanding how coefficients relate to variability and regression analysis.