\begin{question}
For 60 firms the number of employees $X$ and the amount of
expenses for continuing education $Y$ (in EUR) were recorded. The
statistical summary of the data set is given by:
\begin{center}
\begin{tabular}{|l|cc|} \hline
& Variable $X$ & Variable $Y$ \\ \hline
Mean & 56 & 259 \\
Variance & 113 & 1792 \\ \hline
\end{tabular}
\end{center}
The correlation between $X$ and $Y$ is equal to 0.52.
Estimate the expected amount of money spent for continuing education
by a firm with 62 employees using least squares regression.
\end{question}
%% SOLUTIONS
\begin{solution}
First, the regression line $y_i = \beta_0 + \beta_1 x_i +
\varepsilon_i$ is determined. The regression coefficients are given by:
\begin{eqnarray*}
&& \hat \beta_1 = r \cdot \frac{s_y}{s_x} =
0.52 \cdot \sqrt{\frac{1792}{113}} = 2.07078, \\
&& \hat \beta_0 = \bar y - \hat \beta_1 \cdot \bar x =
259 - 2.07078 \cdot 56 = 143.03654.
\end{eqnarray*}
The estimated amount of money spent by a firm with
62 employees is then given by:
\begin{eqnarray*}
\hat y = 143.03654 + 2.07078 \cdot 62 = 271.425.
\end{eqnarray*}
\end{solution}
%% META-INFORMATION
%% \extype{num}
%% \exsolution{271.425}
%% \exname{Prediction}
%% \extol{0.01}