Question

In the graph shown, the solid line represents the best fit from an ordinary least-squares regression, where X is the predictor variable and Y is the response variable. In this case, which one of the following assumptions of the linear regression is violated?

• A. Equal variances in Y across values of X
• B. Normal distribution of residuals
• C. Independence of data points
• D. Linear relationship between X and Y

Hint:

When performing linear regression, check for homoscedasticity (constant variance of residuals). If the spread of residuals increases or decreases with X, it indicates a violation of this assumption.

Answer 1

The Correct Option is A

Step 1: Understanding the Assumptions of Linear Regression.
Linear regression assumes several things about the data, including: 1. A linear relationship between the predictor and response variables. 2. Equal variances in the response variable (homoscedasticity). 3. Normal distribution of residuals. 4. Independence of data points.

Step 2: Analyzing the Graph.
In the provided graph, we see that the spread of the residuals (the vertical distances between the data points and the regression line) varies as X increases. Initially, the residuals appear relatively small but grow larger as X increases, indicating a violation of the assumption of homoscedasticity. This means that the variance of Y is not constant across the values of X. This condition is known as heteroscedasticity, which violates the assumption that the variance of the errors should be constant.

Step 3: Conclusion.
Thus, the assumption that is violated in this case is Equal variances in Y across values of X.

Final Answer: \boxed{(A)}