In simple linear regression with one predictor $x$, what is the equation of the least squares regression line used to predict $y$ from $x$?

• A. $\hat{y}=rx$, where $r$ is the correlation coefficient
• B. $\hat{x}=b+my$, predicting $x$ from $y$
• C. $y=b+mx$, where $m$ and $b$ are chosen so that every data point lies exactly on the line
• D. $\hat{y}=b+mx$, where $m$ is the slope and $b$ is the intercept estimated by minimizing the sum of squared residuals

Answer: (D)

Verified step by step guidance

1. Understand that in simple linear regression, we aim to find a line that best predicts the response variable \(y\) based on the predictor variable \(x\).
2. The general form of the regression line is given by the equation \(\hat{y} = b + m x\), where \(\hat{y}\) is the predicted value of \(y\), \(b\) is the y-intercept, and \(m\) is the slope of the line.
3. The slope \(m\) and intercept \(b\) are not chosen arbitrarily; they are estimated by minimizing the sum of the squared differences between the observed values and the predicted values (these differences are called residuals). This method is known as the least squares criterion.
4. Mathematically, the slope \(m\) can be calculated using the formula \(m = \frac{S_{xy}}{S_{xx}}\), where \(S_{xy}\) is the covariance between \(x\) and \(y\), and \(S_{xx}\) is the variance of \(x\).
5. The intercept \(b\) is then found using the formula \(b = \bar{y} - m \bar{x}\), where \(\bar{x}\) and \(\bar{y}\) are the means of \(x\) and \(y\) respectively.