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:
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 $y_i = \beta_0 + \beta_1 x_i + \varepsilon_i$ is determined. The regression coefficients are given by:
$\begin{eqnarray*}
&& \hat \beta_1 = r \cdot \frac{s_y}{s_x} =
0.52 \cdot \sqrt{\frac{1792}{113}} = 2.07078, \\
&& \hat \beta_0 = \bar y - \hat \beta_1 \cdot \bar x =
259 - 2.07078 \cdot 56 = 143.03654.
\end{eqnarray*}$
The estimated amount of money spent by a firm with 62 employees is then given by:
$\begin{eqnarray*}
\hat y = 143.03654 + 2.07078 \cdot 62 = 271.425.
\end{eqnarray*}$