Questions: HW 4.2: Least-Squares Regression Line Question 2 of 7 (1 point) I Question Attempt: 1 of Unlimited Compute the least-squares regression equation for the given data set. Round the slope and y-intercept to at least four decimal places x 6 3 1 4 5 y 5 3 2 7 1 Regression line equation: ŷ

Solution Steps

To compute the least-squares regression line for the given data set, we need to determine the slope (m) and y-intercept (b) of the line that best fits the data points. This involves calculating the means of the x and y values, the covariance of x and y, and the variance of x. The slope is the covariance divided by the variance, and the y-intercept is calculated using the means and the slope.

To find the least-squares regression line, we first calculate the means of the \(x\) and \(y\) values. The mean of \(x\) is given by:

\[ \bar{x} = \frac{6 + 3 + 1 + 4 + 5}{5} = 3.8 \]

Similarly, the mean of \(y\) is:

\[ \bar{y} = \frac{5 + 3 + 2 + 7 + 1}{5} = 3.6 \]

Next, we calculate the covariance of \(x\) and \(y\) and the variance of \(x\). The covariance is calculated as:

\[ \text{cov}(x, y) = \sum (x_i - \bar{x})(y_i - \bar{y}) = 5.6 \]

The variance of \(x\) is:

\[ \text{var}(x) = \sum (x_i - \bar{x})^2 = 14.8 \]

The slope \(m\) of the regression line is the covariance divided by the variance:

\[ m = \frac{\text{cov}(x, y)}{\text{var}(x)} = \frac{5.6}{14.8} \approx 0.3784 \]

The y-intercept \(b\) is calculated using the means and the slope:

\[ b = \bar{y} - m \cdot \bar{x} = 3.6 - 0.3784 \cdot 3.8 \approx 2.1622 \]

Final Answer