Exam 1

  1. Question

    For 56 firms the number of employees XX and the amount of expenses for continuing education YY (in EUR) were recorded. The statistical summary of the data set is given by:

    Variable XX Variable YY
    Mean 46 220
    Variance 140 1827

    The correlation between XX and YY 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 yi=β0+β1xi+εiy_i = \beta_0 + \beta_1 x_i + \varepsilon_i is determined. The regression coefficients are given by: β̂1=r⋅sysx=0.61⋅1827140=2.20361,β̂0=y‾−β̂1⋅x‾=220−2.20361⋅46=118.63386.\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: ŷ=118.63386+2.20361⋅44=215.593.\begin{eqnarray*} \hat y = 118.63386 + 2.20361 \cdot 44 = 215.593. \end{eqnarray*}