In linear regression using the least squares method, which formula correctly represents the slope of the line of best fit for a set of data points $(x,y)$?

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

Answer: (D)

Verified step by step guidance

1. Recall that in simple linear regression, the slope \( b \) of the line of best fit is calculated using the least squares method, which minimizes the sum of squared residuals between observed and predicted values.
2. Identify the formula for the slope \( b \) as the ratio of the covariance between \( x \) and \( y \) to the variance of \( x \). This can be expressed as: \[ b = \frac{\sum_{i=1}^n (x_i - \overline{x})(y_i - \overline{y})}{\sum_{i=1}^n (x_i - \overline{x})^2} \]
3. Understand that the numerator \( \sum_{i=1}^n (x_i - \overline{x})(y_i - \overline{y}) \) measures how \( x \) and \( y \) vary together (covariance), while the denominator \( \sum_{i=1}^n (x_i - \overline{x})^2 \) measures the variability of \( x \) alone (variance).
4. Note that the other given formulas either incorrectly place \( y \) terms in the denominator or swap numerator and denominator, which does not correspond to the least squares slope formula.
5. Therefore, to find the slope of the regression line, use the formula where the numerator is the sum of the products of deviations of \( x \) and \( y \) from their means, and the denominator is the sum of squared deviations of \( x \) from its mean.