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The solutions to quiz 1 for the statistics 512 spring 2005 course. It includes the computation of the average x, corrected sum of squares of xs (sxx), sum of ys, sum of xy, mean square error (mse), best estimate of variance σ2, regression sum of squares, total sum of squares, standard error of the slope estimate, t-test for testing h0:β=0, degrees of freedom for the t-test, and interpretation of the p-value.

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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

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