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. & AgribusinessA Test of Model UsefulnessTable (13 – 4):One tailed testH : β = 0 1Hoa: β1 < 0, or Ha: β1>Test statistic = t0ˆ β − βS ˆ β1ˆ β − βS ˆ βoTwo tailed testH : β = 0 1Hoo: β1 ≠ 01 1o o1= , to=Test statistic = t1Rejection region t < −tα, or t > tWheret αis based on (n – 2) dfαˆ β1− β1= , tS ˆ β1oˆ βo− βo=S ˆ βRejection region t < −tα/ 2, or t > tα/ 2Wheret αis based on (n – 2) dfFor example (13 – 3), we will choose α = 0.05 and, since n = 5, df = (n – 2) = 3, then therejection region for the two tailed test is:t < −t= − .182, or t > t 3.1820 .02530. 025=os2=SSEn − k=SSEn − 2Where:2SSE ( yi− yˆ ) = SSyy− ˆ β1SS∑=xy∑( y − y)2∑( ∑ yi)2SSyy=i= yi−n2 1.10s = = 0.367 ⇒ s = 0.367 = 0.613We estimated previously 1ˆβ = 0.7, s = 0.61 and SSThus:βt =s / SSxx0.7 0.7= =0.61/ 10 0.19ˆ1 =23.7xx( x )222 ∑ i (15)= ∑ xi− = 55 − = 10n5- 12 -
University of Jordan Agricultural Statistic (605150)Faculty of AgricultureDr. Amer SalmanDept. of Agri. Econ. & AgribusinessRejectionRegionRejectionRegionttab =-3.182ttab =3.182 tcal= 3.7ttab < tcal ; 3.182 < 3.7Since the calculated t value falls in the upper – tail rejection region we reject the nullhypothesis and conclude that the slope of1ˆβ is not 0 and it is significant at 5 % level ofsignificance.Another way to make inference about the slope of1ˆβ is not estimate it using aconfidence interval as follows:ˆ ˆsβ1± tα / 2Sβ1When S βˆ = , and tα / 2is based on (n – k) df.SS xxˆ ˆ⎛ 0.61⎞β1 ± tα / 2Sβ1= 0.7 ± 3.182⎜⎟ = 0.7 ± 0.61⎝ 10 ⎠Thus we estimate with 95 % confidence that the interval from 0.09 to 1.31 includes theslope parameter β1. We would expect a narrower interval if the sample size wereincreased.Test of Goodness of Fit and CorrelationThe closer the observations fall to the regression line (i.e. the smaller the residuals),the greater is the variation in Y “explained” by the estimated regression equation. Thetotal variation in Y is equal to the explained plus the residual variation:- 13 -
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Part 13- Simple linear regression - The University of Jordan

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