Exam 1

  1. Question

    For 60 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 56 259
    Variance 113 1792

    The correlation between XX and YY 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 yi=β0+β1xi+εiy_i = \beta_0 + \beta_1 x_i + \varepsilon_i is determined. The regression coefficients are given by: β^1=rsysx=0.521792113=2.07078,β^0=yβ^1x=2592.0707856=143.03654.\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: y^=143.03654+2.0707862=271.425.\begin{eqnarray*} \hat y = 143.03654 + 2.07078 \cdot 62 = 271.425. \end{eqnarray*}