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.
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: $\begin{eqnarray*} \hat y = 118.63386 + 2.20361 \cdot 44 = 215.593. \end{eqnarray*}$