Part 13- Simple linear regression - The University of Jordan

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University of Jordan Agricultural Statistic (605150)Faculty of AgricultureDr. Amer SalmanDept. of Agri. Econ. & Agribusiness∑22ei1∑×⇒ êi(5) S b =− ∑−×21n k xin k ∑2 bˆ1= 12xiS LSo that S b ˆ o and S ˆb 1are the standard errors of the estimates. SinceU iis normallydistributed, Yiand therefore bˆ o and ˆb 1are also normally distributed, so that we can use thet distribution with n – k degrees of freedom, to test hypotheses about and constantconfidence intervals forbˆ o and ˆb 1.tocalbo− bo= ˆSbo,t1calbˆ1− b=Sb11df = n – k (n – 2 always).α = 5 %; one tailed (α), two tailed (α/2).Example (13 – 6):The following table shows the calculations required to test the statistical significanceofbˆ o and ˆb 1. The value of Yˆ iin the table are obtained by subtracting the values of Xiintothe estimated regression equation Yˆ 2i= 27.12 + 1. 66Xi(The values of yiare obtained bysquaring yi( Y i− Y ).Table (13 – 3):year Y (corn)iXi(fertilizer)Yˆ i( Y − Yˆ)e= i2ei2Xi2xi2yi2( Y i− Y )1 40 6 37.08 2.92 8.5264 36 144 2892 44 10 43.72 0.28 0.0784 100 64 1693 46 12 47.04 -1.04 1.0816 144 36 1214 48 14 50.36 -2.36 5.5696 196 16 815 52 16 53.68 -1.68 2.8224 256 4 256 58 18 57.00 1.00 1.0000 324 0 17 60 22 63.64 -3.64 13.2496 384 16 98 68 24 66.96 1.04 1.0816 576 36 1219 74 26 70.28 3.72 13.8384 676 64 28910 80 32 80.24 -0.24 0.0576 1,024 196 529n=10∑Y i=570Y =57∑ iX =180X =18∑ ie =0∑e2 i=47.3056∑ X2i=3,816∑ x2 i=576∑ y2 i=1,634- 10 -
University of Jordan Agricultural Statistic (605150)Faculty of AgricultureDr. Amer SalmanDept. of Agri. Econ. & AgribusinessS2bˆS bôo∑2ei= ×n − k n∑∑x= 3.92 = 1.982ix2i=47.3056×10 − 2381610(576)= 3.922S bˆ=S bˆ11∑2ei×n − k∑1x= 0.01 ≅ 0.12i47.3056=≅ 0.01(10 − 2)(576)bô− bo27.12 − 0bˆ1− b11.66 − 0Therefore, tocal= = ≅ 13. 7 and t1 = = = 16. 6Sbˆ1.98calSb 0.1Since bothtocaltandot 1 calexceed t tabsignificance we [ t t ]= 2.306 with df = n – k = 8 at 5 % level ofo,1>cal cal tabconclude that bothb 0.025oand b 1are significant at the 5 %level of significance (two tailed).1RejectionRegionAcceptance RegionRejection Regiont =2.306tab α /2t =2.306 13.7 16.6tab α /2- 11 -
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Part 13- Simple linear regression - The University of Jordan

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