Questions: Use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression. x: 8, 3, 12, 9, 5, 10, 13, 6, 7, 11, 4 y: 17.18, 5.24, 20.34, 18.50, 11.08, 19.46, 20.24, 13.46, 15.50, 20.08, 8.34 ŷ = (Round to two decimal places as needed.)

Solution Steps

To find the equation of the regression line, we need to calculate the slope (m) and the y-intercept (b) using the least squares method. The formulas for the slope and y-intercept are:

\[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] \[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \]

Where \( n \) is the number of data points. Once we have \( m \) and \( b \), the regression line equation is \( \hat{y} = mx + b \).

Given the data points: \[ \mathbf{x} = [8, 3, 12, 9, 5, 10, 13, 6, 7, 11, 4] \] \[ \mathbf{y} = [17.18, 5.24, 20.34, 18.50, 11.08, 19.46, 20.24, 13.46, 15.50, 20.08, 8.34] \]

We first calculate the necessary sums: \[ n = 11 \] \[ \sum x = 88 \] \[ \sum y = 169.42 \] \[ \sum xy = 1520.36 \] \[ \sum x^2 = 814 \]

Using the formula for the slope: \[ m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \] Substituting the values: \[ m = \frac{11 \cdot 1520.36 - 88 \cdot 169.42}{11 \cdot 814 - 88^2} \] \[ m = \frac{16723.96 - 14909.96}{8954 - 7744} \] \[ m = \frac{1814}{1210} \] \[ m = 1.5 \]

Using the formula for the y-intercept: \[ b = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \] Substituting the values: \[ b = \frac{169.42 \cdot 814 - 88 \cdot 1520.36}{11 \cdot 814 - 88^2} \] \[ b = \frac{137937.88 - 133791.68}{8954 - 7744} \] \[ b = \frac{4146.20}{1210} \] \[ b = 3.4018 \]

Final Answer