In a two variable regression, the dependent and independent variables are Y and X respectively. The coefficient of correlation between X and Y is 0.8. Which of the following statements is correct?

Correct option is B

Correct Option: B. $$64\%$$​ of variation in $$Y$$​ is explained by $$X$$​​
Explanation:
The topic is the Coefficient of Determination ($$R^2$$​). While the correlation coefficient ($$r$$​) measures the strength and direction of the linear relationship, the coefficient of determination measures the proportion of variation in the dependent variable explained by the independent variable.
Information Booster:
1. Formula: $$R^2 = r \times r$$ (in simple regression).
2. Calculation: Given $$r = 0.8$$​, we calculate $$R^2 = (0.8)^2 = 0.64$$​.
3. Percentage: Converting to percentage, $$0.64 \times 100 = 64\%$$​. This means 64% of the total variation in $$Y$$​ is explained by the regression on $$X.$$​​
Additional Knowledge:
1. Unexplained Variation: The remaining percentage ($$100\% - 64\% = 36\%$$​) is the variation due to the error term (residuals) or other factors not included in the model.
2. Range: $$R^2$$​ always lies between 0 and 1(or 0% to 100%).
3. Relationship: $$r$$ can be negative (e.g., -0.8), but $$R^2$$​ will always be positive because it is a squared value.