If a scatterplot of 1/sqrt(y) vs. x shows a strong positive linear pattern, what can we infer about the residuals plot for the regression of y on x?

If a scatterplot of 1/sqrt(y) vs. x shows a strong positive linear pattern, what can we infer about the residuals plot for the regression of y on x?

• A. The residuals plot will show a linear pattern
• B. The residuals plot for regression of y and x shows a curved pattern
• C. The residuals plot for regression of y and x is irrelevant
• D. The residuals will have no correlation

Answer: (B)

When a scatterplot of 1/sqrt(y) versus x indicates a strong positive linear relationship, this suggests that the transformation applied to y has successfully linearized the relationship between the two variables. In this case, applying the transformation 1/sqrt(y) aligns with a commonly used method in regression analysis to stabilize variance and address potential non-linearity in the original variables. As a result, when performing a linear regression of y on x, the assumptions of linearity and constant variance (homoscedasticity) may be violated. If the transformation indicates a strong linear pattern, the original data may still exhibit non-linear characteristics that would manifest in the residuals plot following the regression of y on x. Consequently, the residuals plot for this regression will likely reveal a curved pattern, indicating that the linear model is not adequately capturing the relationship in the original scale of y. This curved pattern suggests that a different model, possibly involving a transformation or a higher-order polynomial, could be more appropriate for accurately modeling the relationship between x and y.

When a scatterplot of 1/sqrt(y) versus x indicates a strong positive linear relationship, this suggests that the transformation applied to y has successfully linearized the relationship between the two variables. In this case, applying the transformation 1/sqrt(y) aligns with a commonly used method in regression analysis to stabilize variance and address potential non-linearity in the original variables.

As a result, when performing a linear regression of y on x, the assumptions of linearity and constant variance (homoscedasticity) may be violated. If the transformation indicates a strong linear pattern, the original data may still exhibit non-linear characteristics that would manifest in the residuals plot following the regression of y on x.

Consequently, the residuals plot for this regression will likely reveal a curved pattern, indicating that the linear model is not adequately capturing the relationship in the original scale of y. This curved pattern suggests that a different model, possibly involving a transformation or a higher-order polynomial, could be more appropriate for accurately modeling the relationship between x and y.