Question

Suppose we have the following information on the examination scores and weekly employment hours of eight students:

Exam Score      Employment Hours

      80                           10

      80                           5

      68                           15

      95                           5

      75                           21

      60                           40

      90                           0

    100                           0

Assuming Exam Score is the dependent variable (Y) and Employment Hours is the independent variable (X), calculate the simple linear regression equation by hand. Using this, test to see whether the slope on X is significant at the 0.05 level (do this by hand, as well). Make sure to show your work.

Using the model for Question 1, determine the simple linear regression equation using Excel. Using p-values, comment on the goodness of fit for this regression model.

Answer 1

Y: Exam Score = 80, 80,68,95,75,60,50, lo n=no. of Samples = 8 EY= 648 mean=ý = = 81 8 Ey2 = 53774 X: Employment = 10,8,15,5121,40490} Ex = 9G Exi² = 2416 meanX) = EXi = 12 .EKY)= 6670 Simple linear Equations- Y= a + bx X! independent variable Y? Dependent variable a: interrept: b. Slope of line. b= cov (YX) = (xY) E(X).ELY). V(x) E(X- E(X)) = (6679) – (12)-(31) F(x2)-(E(x)32 = 833.75 -972 = -138.25 158 -0.875

Y = a + b Ib= -0.875 a = 9-bx = 81- (-0-875) (12) = 81+10.5 =915 Y=91.5- 0.875 x on X (6) To test to see whether the slope Significat at 0.05 % - To test : b Ho: b=0 against Hiob 70 Test Statistic: = b = b :: YCb) = m Sen Peru (<11993) Sxx 805 49 Y xi x; -x ² i = 91.5-0.875xi Mi-3212Lxi-x) 80 10 4 82.75 7.5625 87.925 150.76561 68 159 78.375 107.84 87. 125 62.0156 752181 73.125 3.51562 60 60 784 56.5 12.25 1164 2.25 648 1318.24931 915 915 72.25 144 1269

- 0. 0.05196 VI VC) = (318-2493) = 53.on155 11265 318 - 2493 8-2 1264 TV 6) = 0.204848 ; 2 = b-b = -0.875-0 0. 204848 – – 4.27145 = -0.875 0. 204848 Zeal = -4.27145 it Zeal) = 4.27145 21 = Zoros = Zo.ois 1.96. Leal Decision & Reject Ho when Zcal 17 Zx 12 - Hence, Ho is significant So & Reject Ho Hereg (Zew) = 4.27145) 7 (20.025 1.96) & Conclusion: b is significant je': Sope on X je- significa cet 0.05 l'O's!