Find the Least-Squares Regression Line and Use the Line to Make Predictions
-A manager wishes to determine the relationship between the number of years her sales representatives have been employed by the firm and their amount of sales (in thousands of dollars) per month. Find the equation of the regression line for the given data. What would be the predicted sales if the sales representative was employed by the firm for 30 years Is this reasonable? Why or why not?
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline \text { Years employed, x } & 2 & 3 & 10 & 7 & 8 & 15 & 3 & 1 & 11 \
\hline \text { Sales, y } & 31 & 33 & 78 & 62 & 65 & 61 & 48 & 55 & 120
\end{array}$

A) $\hat { y } = 3.53 \mathrm { x } + 37.92 ; \$ 143,820$; No; it is not reasonable. 30 years of employment is well outside the scope of the model.
B) $\hat { y } = 3.53 x + 37.92 ; \$ 143,820 ;$ Yes, it is reasonable.
C) $\hat { y } = 3.53 \mathrm { x } - 37.92 ; \$ 67,980 ; \mathrm { No }$; it is not reasonable. 30 years of employment is well outside the scope of the model.
D) $\hat { y } = 3.53 \mathrm { x } - 37.92 ; \$ 67,980$; Yes; it is reasonable.

Question

Find the Least-Squares Regression Line and Use the Line to Make Predictions
-A manager wishes to determine the relationship between the number of years her sales representatives have been employed by the firm and their amount of sales (in thousands of dollars) per month. Find the equation of the regression line for the given data. What would be the predicted sales if the sales representative was employed by the firm for 30 years Is this reasonable? Why or why not?
$\begin{array}{|l|c|c|c|c|c|c|c|c|c|}
\hline \text { Years employed, x } & 2 & 3 & 10 & 7 & 8 & 15 & 3 & 1 & 11 \
\hline \text { Sales, y } & 31 & 33 & 78 & 62 & 65 & 61 & 48 & 55 & 120
\end{array}$

A) $\hat { y } = 3.53 \mathrm { x } + 37.92 ; \$ 143,820$; No; it is not reasonable. 30 years of employment is well outside the scope of the model.
B) $\hat { y } = 3.53 x + 37.92 ; \$ 143,820 ;$ Yes, it is reasonable.
C) $\hat  { y } = 3.53 \mathrm { x } - 37.92 ; \$ 67,980 ; \mathrm { No }$; it is not reasonable. 30 years of employment is well outside the scope of the model.
D) $\hat  { y } = 3.53 \mathrm { x } - 37.92 ; \$ 67,980$; Yes; it is reasonable.

Quizplus · Accepted Answer

The correct equation for the least-squares regression line cannot be determined without performing the actual calculations involving the given data points. However, the prediction for a sales representative employed for 30 years and the reasoning about its reasonableness can be addressed conceptually. Predictions made for values (like 30 years of employment) that are far outside the range of the data used to create the model (1 to 15 years, in this case) are generally not considered reliable. This is because the model is extrapolating far beyond the data it was built upon, and there's no guarantee that the observed relationship holds outside that range.