Question

I just need help with questions 5,6 and 7

Economists wish to look at the relationship between the price of regular unleaded gasoline in the U.S. (units-$/gallon) and the acquisition cost of OPEC crude oil (units-$10/barrel). Summary measures for a random sample of 13 years are given below 13 29.3 0.39 329.67 143.61 What is the estimated regression equation? (Specify which variable is y and which is x) Question 1 where ice of gas t of Question 2 Show the ANOVA table MS dt Regression Error Total

Answer 1

$Q1.~\bar{x}=\frac{1}{13}\sum x_i=4.6538,~\bar{y}=\frac{1}{13}\sum y_i=2.2538,\\ ~S_{xy}=\sum x_iy_i-13\bar{x}\bar{y}= 7.2565,~S_{xx}=\sum x_i^2-13\bar{x}^2=48.1179,\\ b=estimate~of~slope~(\beta)=\frac{S_{xy}}{S_{xx}}= 0.1508,\\ a=estimate~of~intercept~(\alpha)=\bar{y}-b\bar{x}=1.5520,\\ \hat{y}=1.5520+0.1508x\\ Question~2:\\ SST=S_{yy}=sum~of~square~due~to~total=\sum y_i^2-13\bar{y}^2=4.3550.\\ r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=0.5013.\\ SSR=sum~of~square~due~to~regression=SST\times r^2=1.0944~with~df=1,\\ MSR=SSR/1=1.0944,\\ SSE=4.3550-1.0944=3.2606~with~df=11\\ MSE=SSE/11=0.2964,\\ \begin{matrix} & df &SS &MS \\ \hline Regression &1 & 1.0944 &1.0944 \\ Error & 11 & 3.2606 &0.2964 \\ Total&12 &4.3550 & \\ \hline \end{matrix}\\ Q3.~r^2=0.5013^2=0.2513.\\ Q4:\\ Lower~End point=b-t_{0.05,11}\times \sqrt{\frac{MSE}{S_{xx}}}=0.0098\\ where,~t_{0.05,11}=1.7959\\ Upper~End~point=b+t_{0.05,11}\times \sqrt{\frac{MSE}{S_{xx}}}=0.2918\\$

$Q5.~F_{crit}=F_{0.025,1,11}=6.7241 ~F-stat=MSR/MSE.\\ Reject~H_0~if~F~stat>F_{crit}.\\ F-stat=1.0944/0.2964=3.6923\\ We~fail~to~reject~H_0.\\ p-value=P(F>3.6923|F\sim F_{1,11})=0.0809.\\ 0.05<p-value<0.1.\\ Question~6:\\ y_0=1.5520&plus;0.1508*2.9=1.9893\\ Lower~end point=y_0-t_{0.025,11}\sqrt{MSE\left(\frac{1}{13}&plus;\frac{(2.9-\bar{x})^2}{S_{xx}} \right )}=1.5396\\ where,~t_{0.025,11}=2.2010\\ Upper~end point=y_0&plus;t_{0.025,11}\sqrt{MSE\left(\frac{1}{13}&plus;\frac{(2.9-\bar{x})^2}{S_{xx}} \right )}=2.4390\\ Q7.a.~Predicted~value=y_0=1.5520&plus;0.1508*1.7=1.8084\\ b.\\ Lower~end point=y_0-t_{0.1,11}\sqrt{MSE\left(1&plus;\frac{1}{13}&plus;\frac{(1.7-\bar{x})^2}{S_{xx}} \right )}=1.5396\\ where,~t_{0.1,11}=0.9758\\ Upper~end point=y_0&plus;t_{0.1,11}\sqrt{MSE\left(1&plus;\frac{1}{13}&plus;\frac{(1.7-\bar{x})^2}{S_{xx}} \right )}=2.6410\\$