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. $$ \begin{array}{lrrrr} \text { Coefficients: } & \text { Estimate } & \text { Std.Error } & \mathrm{t} \text { value } & \mathrm{Pr}(>|\mathrm{t}|) \\ \text { (Intercept) } & 7.277 & 1.167 & 6.24 & 0.000 \\ \text { Dose } & -0.3560 & 0.2007 & -1.77 & 0.087 \end{array} $$

Short answer

The intercept (\(\beta_0\)) is 7.277, the slope (\(\beta_1\)) is -0.3560, and the equation of the least squares line is \(\hat{Y} = 7.277 - 0.3560* Dose\).

Step-by-step solution

Identify the Intercept (\(\beta_0\))

The Intercept (\(\beta_0\)) is given in the output table under the 'Estimate' column next to '(Intercept)'. From the table, \(\beta_0 = 7.277\). This value represents the estimated value of \(Y\) when all other predictor variables are zero.

Identify the Slope (\(\beta_1\))

The Slope (\(\beta_1\)) is given in the output table under the 'Estimate' column next to 'Dose'. From the table, \(\beta_1 = -0.3560\). This value represents how much \(Y\) changes for each one-unit change in the predictor variable 'Dose'.

Formulate the Least Squares Line

The equation for the least squares line is always of the form \(\hat{Y} = \beta_0 + \beta_1X\). Based on the estimated values of \(\beta_0\) and \(\beta_1\) obtained in Step 1 and Step 2, the equation of the least squares line can be written as \(\hat{Y} = 7.277 - 0.3560* Dose\).