I was hoping someone might explain why answer C in the picture is better than D or E? It would appear that in answer D, the larger x becomes the closer the regression line fits the data. With E, the smaller x is the better the regression line fits.
Thanks a lot!
The regression model $Y_i=β_0+β_1 x_i+e_i$ assumes $e_i\stackrel{iid}{\sim}\mathsf{Norm}(0,σ),$ which implies that $σ$ is the same for all $x_i.$ In 'regression diagnostics', residuals provide the best clue as to the behavior of the $e_i.$
(c) Shows stable variance of residuals across values of $x.$ There is no pattern among residuals, this suggests that "errors" are random.
In (e) residuals 'fan out' with more variable residuals as $x$ increases.
In (d) residuals become less variable with increasing $x.$
So residual plots (d) and (e) suggest a violation of the constant-variance assumption of the regression model. If variances are not constant, then the t tests of $H_0: \beta_0 = 0$ vs. $H_a: \beta_0 \ne 0$ and of $H_0: \beta_1 = 0$ vs. $H_a: \beta_1 \ne 0$ may not be valid, and prediction intervals for $\hat Y_{n+1}$ corresponding to a new $x_{n+1}$ cannot be trusted.