5/3/00 252y0043 (Open this document in 'Page Layout' view!) ECO252 QBA2 Name key FINAL EXAM May 4, 2000 Note: If this is the only thing you look at before taking the final, you are badly cheating yourself. People who used last year's final and did not read the problems carefully got very wrong answers to them. If you can't be bothered to think, there is not much point to taking this course or this exam.Note: If you still think that a large p-value means that a coefficient is significant, you need a conference with an audiologist. Further note that a p-value is a probability and can only be compared with another probability (like the significance level).Note: Have you reread “Things that You Should Never Do On a Statistics Exam …?” I think I could have graded this exam by just looking for violations of these rules.I. (16 points) Do all the following.1. Hand in your fourth regression problem (2 points) and answer the following questions.a. For the regression of revenue against 'gdp' and 'gdpsq' (gdp squared), what coefficients are significant at the 5% level? Why? What about the 10% level? (2)b. In the same regression, which coefficients have signs that are/are not reasonable? Explain why. (1)c. Do an F-test to show if addition of both 'yearsq' and 'gdp' improves the regression over your results with 'year' alone? (4)d. Overall, which of the regressions seems most successful in predicting 'rev'? Why? (3)2. The following pages show the regression of the variable 'time', the time in minutes it takes to do a task on an automobile assembly line, against months of experience 'exp':The intent was to fit the model, 332210xxxy where xis months of experience. Accordingly, 'exp,' 'expsq' and 'expcu' represent , respectively, the months of experience of a given worker, experience squared and experience to the third power.a. What is our equation that relates ‘time’ against 'exp,' 'expsq' and 'expcu' ? What time does it predict for someone with experience of 12 months (Show your work!) (2) b. Look at these equations and do appropriate tests 05.to show whether we are justified in including experience squared or cubed in the equation. (4) Solution: 1) a. The result from the computer output (see document 252cp400s) is copied below. MTB > regress 'rev' on 2 'gdp''gdpsq''resid''pred'Regression AnalysisThe regression equation isrev = 65.7 - 26.5 gdp + 2.88 gdpsqPredictor Coef Stdev t-ratio pConstant 65.67 67.70 0.97 0.360gdp -26.54 21.92 -1.21 0.261gdpsq 2.877 1.769 1.63 0.143s = 1.085 R-sq = 94.5% R-sq(adj) = 93.1%Since all these p-values are above both 5% and 10%, none of them are significant at either level.The rule on p-value: If the p-value is less than the significance level reject the nullhypothesis; if the p-value is greater or equal than the significance level, do not reject the null hypothesis.5/3/00 252y0043b. We would expect sales to go up with gdp, a negative coefficient doesn't look right. There is nothing wrong with a negative regression coefficient, unless you have a good reason to believe that it shouldn't be negative.c. The result from the computer output (see document 252cp400s) is copied below.MTB > regress 'rev' on 3 'year''yearsq''gdp''resid''pred'Regression AnalysisThe regression equation isrev = 16.7 + 0.870 year + 0.0588 yearsq - 1.77 gdpPredictor Coef Stdev t-ratio pConstant 16.705 7.684 2.17 0.066year 0.8698 0.2159 4.03 0.005yearsq 0.05882 0.01301 4.52 0.000gdp -1.773 1.382 -1.28 0.240s = 0.3800 R-sq = 99.4% R-sq(adj) = 99.2%Analysis of VarianceSOURCE DF SS MS F pRegression 3 169.255 56.418 390.81 0.000Error 7 1.011 0.144Total 10 170.265SOURCE DF SEQ SSyear 1 166.173yearsq 1 2.844gdp 1 0.238If we look at the sequential sums of squares at the end of the ANOVA, we see that 'yearsq' and 'gdp' together account for 2.844 + 0.238 = 3.082. We can use this to itemize the ANOVA as follows:SOURCE DF SS MS F pyear 1 166.173 166.173yearsq, gdp 2 3.082 1.541 10.701Error 7 1.011 0.144Total 10 170.265Since 74.47,205.F, we can conclude that the addition of these two variables improves the regression, though a look at the p-values for the coefficients makes it look like this improvement is due to 'yearsq' alone.d. If our criteria are a high 2R and significant (low p-value) coefficients, we don’t seem to be able to improve onrev = 6.85 + 0.653 year + 0.0576 yearsqPredictor Coef Stdev t-ratio pConstant 6.8545 0.3009 22.78 0.000year 0.6533 0.1400 4.67 0.000yearsq 0.05758 0.01348 4.27 0.003s = 0.3950 R-sq = 99.3% R-sq(adj) = 99.1%Any higher 2R seems to be accompanied by insignificant coefficients and often unreasonable signs on these coefficients. In particular, the negative, insignificant coefficient of gdp in the equation toward the top of this page disqualifies it. This is what STEPWISE should have made you suspect.2) See below. 25/3/00 252y0043Worksheet size: 100000 cellsMTB > RETR 'C:\MINITAB\2X0041-1.MTW'.Retrieving worksheet from file: C:\MINITAB\2X0041-1.MTWWorksheet was saved on 5/ 1/2000MTB > regress 'time' on 1 'exp'Regression AnalysisThe regression equation istime = 19.7 - 0.441 expPredictor Coef Stdev t-ratio pConstant 19.7488 0.5407 36.52 0.000exp -0.44116 0.04098 -10.77 0.000s = 1.141 R-sq = 89.9% R-sq(adj) = 89.1%Analysis of VarianceSOURCE DF SS MS F pRegression 1 150.82 150.82 115.90 0.000Error 13 16.92 1.30Total 14 167.73Unusual ObservationsObs. exp time Fit Stdev.Fit Residual St.Resid 1 25.0 11.000 8.720 0.642 2.280 2.42R R denotes an obs. with a large st. resid.MTB > regress 'time' on 2 'exp''expsq'Regression AnalysisThe regression equation istime = 20.8 - 0.636 exp +0.000331
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