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:
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{}& \hfill & {\hat{\mathit{\beta }}}_1=r·\frac{s_y}{s_x}=0.61·\sqrt{\frac{1827}{140}}=2.20361,\hfill \\ \multicolumn{1}{c}{}& \hfill & {\hat{\mathit{\beta }}}_0=\bar{y}-{\hat{\mathit{\beta }}}_1·\bar{x}=220-2.20361·46=118.63386.\hfill \end{array}}$

   
   
   The estimated amount of money spent by a firm with 44 employees is then given by:
   

   ${\displaystyle \begin{array}{c}\multicolumn{1}{c}{\hat{y}=118.63386+2.20361·44=215.593.}\end{array}}$