## Exam 1

1. #### 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:
 Variable $X$ Variable $Y$ Mean 56 259 Variance 113 1792
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.

#### Solution

First, the regression line ${y}_{i}={\beta }_{0}+{\beta }_{1}{x}_{i}+{ϵ}_{i}$ is determined. The regression coefficients are given by:
 $\begin{array}{ccc}\multicolumn{1}{c}{}& \hfill & {\stackrel{^}{\beta }}_{1}=r·\frac{{s}_{y}}{{s}_{x}}=0.52·\sqrt{\frac{1792}{113}}=2.07078,\hfill \\ \multicolumn{1}{c}{}& \hfill & {\stackrel{^}{\beta }}_{0}=\stackrel{‾}{y}-{\stackrel{^}{\beta }}_{1}·\stackrel{‾}{x}=259-2.07078·56=143.03654.\hfill \end{array}$

The estimated amount of money spent by a firm with 62 employees is then given by:
 $\begin{array}{c}\multicolumn{1}{c}{\stackrel{^}{y}=143.03654+2.07078·62=271.425.}\end{array}$