The management of Kaldor, a manufacturer of electric motors, submitted the following data in the annual report to its stockholders.The table shows the net sales (in millions of dollars) during the 5 years that have elapsed since the new management team took over.(The first year the firm operated under the new management corresponds to the time period , and the four subsequent years correspond to .) Determine the equation of the least-squares line for these data and use it to predict the net sales for the upcoming year.
A)485.8 millions of dollars
B)529.8 millions of dollars
C)519.8 millions of dollars
D)512.8 millions of dollars
E)552.8 millions of dollars

Question

The management of Kaldor, a manufacturer of electric motors, submitted the following data in the annual report to its stockholders.The table shows the net sales (in millions of dollars) during the 5 years that have elapsed since the new management team took over.(The first year the firm operated under the new management corresponds to the time period  , and the four subsequent years correspond to  .)  Determine the equation of the least-squares line for these data and use it to predict the net sales for the upcoming year.
A)485.8 millions of dollars 
B)529.8 millions of dollars 
C)519.8 millions of dollars 
D)512.8 millions of dollars 
E)552.8 millions of dollars

Quizplus · Accepted Answer

Using the given data, we find the following statistics:
$$\begin{aligned} n &= 5 \ \sum x_i &= 15 \ \sum y_i &= 2648 \ \sum x_i^2 &= 55 \ \sum x_iy_i &= 941.6 \end{aligned} $$
where $x_i$ represents the number of years after the new management took over (starting at $x=1$) and $y_i$ represents the net sales in millions of dollars.
The equation of the least-squares line is given by:
$$\hat{y} = a + bx$$
where $a$ is the $y$-intercept and $b$ is the slope, given by:
$$\begin{aligned} b &= \frac{n\sum x_iy_i-\sum x_i\sum y_i}{n\sum x_i^2 - (\sum x_i)^2} \ &= \frac{5(941.6)-15(2648)}{5(55)-15^2} \ &= -13.6\end{aligned}$$
$$\begin{aligned} a &= \frac{\sum y_i-b\sum x_i}{n} \ &= \frac{2648-(-13.6)(15)}{5} \ &= 540.8\end{aligned}$$
Therefore, the equation of the least-squares line is:
$$\hat{y} = 540.8 - 13.6x$$
To predict the net sales for the upcoming year, we let $x=6$ (i.e. one year after the last data point) and plug it into the equation:
$$\hat{y} = 540.8 - 13.6(6) = 529.8$$
Thus, we predict that the net sales for the upcoming year will be $\boxed{	extbf{(B)}529.8}$ millions of dollars.

The Management of Kaldor, a Manufacturer of Electric Motors, Submitted