Download Regression Analysis Quiz Solution for St 512 Spring 2005 - Prof. David Dickey and more Quizzes Statistics in PDF only on Docsity! Quiz 1 St 512 Spring 2005 Dickey A line is fitted to 10 data points (Xi, Yi). The equation of this fitted line is Y=10 + 0.3X and the population model that is being estimated is Y = α + βX + e where the errors e have the usual assumptions, namely e ~N(0,σ2) and the e’s are independent of each other. The values of X are 10, 20, 30, 40, 50 and there are two points above each X giving us our 10 total points. I also computed the sum of the squared residuals (vertical deviations of the points from the line) getting 640 as my error sum of squares. 1. (16 pts.) Compute the average _____ of the 10 X values and the (corrected) sum of squares of the Xs. Sxx=_________ 2. (8 pts.) Compute ∑ = =−− 10 1 _______))(( i ii YYXX (If you can’t do it, insert a guess and use it later when needed) 3. (10 pts.) Compute the error mean square for this regression. MSE = ________ 4. (8 pts.) We assumed the e’s have variance σ2. Give your best estimate of σ2 _________ 5. (16 pts.) Compute, if possible, the regression sum of squares ______ and the (corrected) total sum of squares Syy=_________ for these data. 6. (24 pts.) Our slope estimate, 0.3, has a standard error. Compute that standard error _____ and the t test _____for testing H0:β=0. How many degrees of freedom _____ does this t have? 7. (8 pts.) Assuming the p-value for the test in 6 is p=0.1720, do you reject or fail to reject the null hypothesis using the usual 5% level of significance? 8. (10 pts.) Insert the three labels in the picture below to explain what the p-value in question 7 represents. Total shaded area is ___.1720______ | | (___-1.5___) (__1.5___) **************MORE ANSWERS ********************************* It is easy to see that the mean of X is 30, the deviations are -20, -10, 0, 10, 20 each occurring twice so squaring and summing the 10 deviations we get Sxx=2000. Since b is Sxy/Sxx we find Sxy=(2000)(b) = (2000)(0.3)= 600 * (see note below) MSE = 640/8 = 80 80 is ALSO best estimate σ2 SS regression is 600*600/2000 = 360000/2000= 260/2 = 180. SS total = SS regression + SS error = 180 + 640 = 820. 2.02000/80/ ==SxxMSE t = 0.3/0.2 = 1.5 with 8 df. 0.1720 > 0.0500 so we fail to reject H0. ----------------------------------------------------------------------------------------------- * Note that Sxy would be computed from the data as ∑ −− ))(( YYXX ii . For the observed points (Xi, Yi) used in this formula, Yi is not the predicted value 10+0.3X. It is only for points Y on the line that Y=10+0.3X. Therefore, you would think that substituting the predicted values 10+0.3X for Yi in the Sxy formula would give the wrong answer. Indeed every term in that sum will change, but the sum of them will
