## Exam 1

1. #### 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:
 Variable $X$ Variable $Y$ Mean 46 220 Variance 140 1827
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.

#### Solution

First, the regression line ${y}_{i}={\mathit{\beta }}_{0}+{\mathit{\beta }}_{1}{x}_{i}+{\mathit{ϵ}}_{i}$ is determined. The regression coefficients are given by:
 $\begin{array}{ccc}\multicolumn{1}{c}{}& \hfill & {\stackrel{^}{\mathit{\beta }}}_{1}=r·\frac{{s}_{y}}{{s}_{x}}=0.61·\sqrt{\frac{1827}{140}}=2.20361,\hfill \\ \multicolumn{1}{c}{}& \hfill & {\stackrel{^}{\mathit{\beta }}}_{0}=\stackrel{‾}{y}-{\stackrel{^}{\mathit{\beta }}}_{1}·\stackrel{‾}{x}=220-2.20361·46=118.63386.\hfill \end{array}$

The estimated amount of money spent by a firm with 44 employees is then given by:
 $\begin{array}{c}\multicolumn{1}{c}{\stackrel{^}{y}=118.63386+2.20361·44=215.593.}\end{array}$