# Part d 15 pts although not a very useful or common

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Part D (15 pts)Although not a very useful or common approach, we now consider a testing procedure to seeif the intercept statistically diers from the slope, i.e., consider testing the null/alternativepairH0:β0=β1versusHA:β06=β1.Write downboththe T-statistic and F-statistic for testing the above null/alternative pair.When constructing the F-statistic, also identify the full and reduced models.Write yoursolution on pages 4 and 5.Note: when specifying the full and reduced models, you do not have to derivethe maximum likelihood estimators but make sure to identify them.4
5F-testFIY;=po+p,xite;,e.intNato?)ML-Estimators:§,=55×4,do=F-B.ISSEF=If(Y;-pro-§,x:)2=SSEdff=n-2ReducedUnderHo:p.=p,Y;=p,+p,Xi+E;=B.(ITXI)+E,eiiitvlo,o2)ML-Estimator§=ICXITDYIZCX;+1)2SSER=E.(Y;-Fx;)2dfr=n-1F-statisticF=(SSER-SsE)11SEF~f(df,=1,dfz=n-2)
Part E (15 pts)Consider the following toy dataset displayed in the scatter plot below.Let the predictorvariable be assigned asx, the response asYand assignnas the sample size. Note that therearen= 100 cases in this dataset.Using theRcode and output displayed on pages 6, 7 and 8, test the if the intercept statis-tically diers from the slope, i.e., test the null/alternative pair:H0:β0=β1versusHA:β06=β1.To receive full credit, computeboththe T-statistic and F-statistic for testing the abovenull/alternative pair. Also compute the correct p-value and state the statistical conclusion.Write the solution on the top of page 7.Note:1-pt(t.calc,98)=0.11852181-pf(f.calc,1,98)=0.23704371-pt(t.calc,99)=0.11850741-pf(f.calc,1,99)=0.23701476
> # Model 1 with Summary and ANOVA #-----------------------------------------------> summary(lm(Y~x))Coefficients:Estimate Std. Error t value Pr(>|t|)(Intercept)1.17080.31443.725 0.000327 ***x0.73980.054113.676< 2e-16 ***Residual standard error: 1.563 on 98 degrees of freedomMultiple R-squared:0.6562,Adjusted R-squared:0.6527F-statistic:187 on 1 and 98 DF,p-value: < 2.2e-16> anova(lm(Y~x))Df Sum Sq Mean Sq F valuePr(>F)x
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