In simple linear regression using the least squares method, which plot will produce a straight line if the model assumptions are satisfied?

• A. A plot of the predicted values of $y$ versus the observed values of $x$
• B. A plot of the residuals versus the predicted values of $y$
• C. A plot of the residuals versus the observed values of $x$
• D. A plot of the observed values of $y$ versus the observed values of $x$

Answer: (A)

Verified step by step guidance

1. Recall that in simple linear regression, the model assumes a linear relationship between the independent variable \(x\) and the dependent variable \(y\), expressed as \(y = \beta_0 + \beta_1 x + \varepsilon\), where \(\varepsilon\) is the error term.
2. Understand that the predicted values \(\hat{y}\) are calculated from the estimated regression line: \(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\).
3. Consider the different plots mentioned: plotting \(\hat{y}\) versus \(x\) should show a straight line if the model assumptions hold, because \(\hat{y}\) is a linear function of \(x\).
4. Plots of residuals (the differences between observed and predicted values) versus predicted values or versus \(x\) are used to check model assumptions like homoscedasticity and independence, but these plots do not produce a straight line.
5. Therefore, the plot that will produce a straight line under the model assumptions is the plot of predicted values \(\hat{y}\) versus the observed values of \(x\).