In linear regression using the least squares method, which expression can be simplified to find the slope of the trend line in the scatterplot?

• A. $\frac{\sum _{i=1}(y{i}_{}-\bar{y}{)}^2}{\sum _{i=1}(x{i}_{}-\bar{x}{)}^2}$
• B. $\frac{\sum _{i=1}(x{i}_{}-\bar{x})}{\sum _{i=1}(y{i}_{}-\bar{y})}$
• C. $\frac{\sum _{i=1}(x{i}_{}-\bar{x})(y{i}_{}-\bar{y})}{\sum _{i=1}(x{i}_{}-\bar{x}{)}^2}$
• D. $\frac{\sum _{i=1}(x{i}_{}-\bar{x}{)}^2}{\sum _{i=1}(y{i}_{}-\bar{y}{)}^2}$

Answer: (C)

Verified step by step guidance

1. Recall that in simple linear regression, the slope \( b_1 \) of the least squares regression line is given by the formula involving sums of deviations from the means of \( x \) and \( y \).
2. Write down the formula for the slope \( b_1 \) as \( b_1 = \frac{\sum_{i=1}^n (x_i - \overline{x})(y_i - \overline{y})}{\sum_{i=1}^n (x_i - \overline{x})^2} \), where \( \overline{x} \) and \( \overline{y} \) are the sample means of \( x \) and \( y \) respectively.
3. Understand that the numerator \( \sum (x_i - \overline{x})(y_i - \overline{y}) \) represents the covariance between \( x \) and \( y \), measuring how they vary together.
4. Recognize that the denominator \( \sum (x_i - \overline{x})^2 \) is the variance of \( x \), measuring the spread of the \( x \)-values around their mean.
5. Therefore, the expression that can be simplified to find the slope of the trend line is the ratio of the covariance of \( x \) and \( y \) to the variance of \( x \), which matches the formula \( b_1 = \frac{\sum_{i=1}^n (x_i - \overline{x})(y_i - \overline{y})}{\sum_{i=1}^n (x_i - \overline{x})^2} \).