Given the least squares regression equation $y=a+bx$, what does the coefficient $b$ represent?

• A. The correlation coefficient between $x$ and $y$
• B. The value of $y$ when $x=0$
• C. The expected change in $y$ for a one-unit increase in $x$
• D. The total sum of squared residuals

Answer: (C)

Verified step by step guidance

1. Recall the general form of the least squares regression equation: \(y = a + b x\), where \(a\) is the intercept and \(b\) is the slope coefficient.
2. Understand that the coefficient \(b\) quantifies the relationship between the independent variable \(x\) and the dependent variable \(y\).
3. Interpret \(b\) as the expected change in the value of \(y\) when \(x\) increases by one unit, holding all else constant.
4. Note that \(b\) is not the correlation coefficient; the correlation coefficient measures the strength and direction of a linear relationship but is a separate statistic.
5. Also recognize that \(a\) represents the expected value of \(y\) when \(x = 0\), and the total sum of squared residuals is a measure of the overall fit, not a coefficient.