Question

Use the computer output (from different computer packages) to estimate the intercept \(\beta_{0},\) the slope \(\beta_{1},\) and to give the equation for the least squares line for the sample. Assume the response variable is \(Y\) in each case. $$ \begin{array}{lrrrr} \text { Coefficients: } & \text { Estimate } & \text { Std.Error } & \mathrm{t} \text { value } & \operatorname{Pr}(>|\mathrm{t}|) \\ \text { (Intercept) } & 77.44 & 14.43 & 5.37 & 0.000 \\ \text { Score } & -15.904 & 5.721 & -2.78 & 0.012 \end{array} $$

Short answer

The intercept, \(\beta_{0}\), is 77.44, the slope, \(\beta_{1}\), is -15.904, and the equation for the least squares line is \(Y = 77.44 - 15.904*Score\).

Step-by-step solution

Identify Intercept (beta_0)

The intercept, or \(\beta_{0}\), is represented in the 'Estimate' column on the '(Intercept)' row, it has a value of 77.44.

Identify Slope (beta_1)

The slope, or \(\beta_{1}\), is represented in the 'Estimate' column on the 'Score' row, it has a value of -15.904.

Formulate the Least Squares Line

With the slope and intercept identified, you can form the equation for the least squares line. It generally has the form \(\beta_{0} + \beta_{1}X\), or in this case \(Y = \beta_{0} + \beta_{1}*Score\). Substituting the determined values, it will be \(Y = 77.44 - 15.904*Score\).