
\begin{question}
For 56 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}{lcc} \hline
& Variable $X$ & Variable $Y$ \\ \hline
Mean & 46  & 220  \\
Variance & 140  & 1827 \\ \hline
\end{tabular}
\end{center}

The correlation between $X$ and $Y$ is equal to 0.61.

Estimate the expected amount of money spent for continuing education
by a firm with 44 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} = 
0.61 \cdot \sqrt{\frac{1827}{140}} = 2.20361, \\
&& \hat \beta_0 = \bar y - \hat \beta_1 \cdot \bar x = 
220 - 2.20361 \cdot 46 = 118.63386.
\end{eqnarray*}

The estimated amount of money spent by a firm with
44 employees is then given by:
\begin{eqnarray*}
\hat y = 118.63386 + 2.20361 \cdot 44 = 215.593.
\end{eqnarray*}
\end{solution}

%% META-INFORMATION
%% \extype{num}
%% \exsolution{215.593}
%% \exname{Prediction}
%% \extol{0.01}
