\(\displaystyle{P}_{{{W}{T}}}\ {R}_{{{e}{g}}}\ {L}_{{{1}}},\ {L}_{{{2}}}\) Finally, pressing on ENTER then gives us the following result: \(\displaystyle{y}={a}\ \cdot\ {x}^{{{b}}}\)

\(\displaystyle{a}={1.003}\)

\(\displaystyle{b}={1.4998}\)

\(\displaystyle{r}^{{{2}}}={1}\) This then implies that the regression line is: \(\displaystyle\hat{{{y}}}={a}\ \cdot\ {x}^{{{b}}}={1.0003}\ \cdot\ {x}^{{{1.4998}}}\) where x represents the distance from sun and y represents the period of revolution. Step 2 Given: \(\displaystyle{x}={30.061}\)

\(\displaystyle{y}={164.810}\) Evaluate the equation of the regression line at \(\displaystyle{x}={30.061}:\)

\(\displaystyle\hat{{{y}}}={1.0003}\ \cdot\ {30.061}^{{{1.4998}}}\ \approx\ {164.7631}\) The residual is the difference between the observed y-value and the predicted y-value. \(\displaystyle\text{Residual}\ ={y}\ -\ \hat{{{y}}}\ ={164.810}\ -\ {164.7631}={0.0469}\) Neptune's period of revolution is 0.0469 earth years longer than expected.