SourceSSDFMSSourceSSDFMS4/26/02 252y0232 ECO252 QBA2 Name KEY (Page layout view!) THIRD HOUR EXAM Hour of Class Registered (Circle) April 18, 2002 MWF TR 10 12 12:30 2:00I. (10+ points) Do all the following;1. Hand in your computer printouts for problems 2 and 3.(5 points – 3 point penalty for not handing in). remember that the ANOVA printout must be completed, using a 5% significance level, for full credit. I should be able to tell what is tested and what are the conclusions.2. a. In particular, is the interaction between car and driver significant? Which numbers made you think that? (2) b. Create two confidence intervals for the difference between the means for drivers 2 and 3, one that is valid alone, and one that is valid simultaneously with other similar intervals. Do these intervals show a significant difference between these two means? Why? (4) Solution: The only parts of the solution to computer problem 2 that you need are:Tabulated Statistics ROWS: car COLUMNS: driver 1 2 3 ALL 1 42.000 25.000 12.667 26.556 2 32.000 28.000 29.333 29.778 3 30.667 45.000 28.333 34.667 4 31.333 24.667 54.667 36.889 ALL 34.000 30.667 31.250 31.972 CELL CONTENTS -- mpg:MEANMTB > twoway 'mpg''car''driver'Two-way Analysis of VarianceAnalysis of Variance for mpg Source DF SS MScar 3 590.3 196.8driver 2 76.1 38.0Interaction 6 3227.9 538.0Error 24 336.7 14.0Total 35 4231.0To complete the printout, divide through the MS column by 14MSW and place the results in the in the F column. Then look up the corresponding values of F in 5% lines on the F table.Source DF SS MS F 05.F 0Hcar 3 590.3 196.8 14.057s 01.324,305.F Car means identicaldriver 2 76.1 38.0 2.714ns 40.324,205.F Driver means identicalInteraction 6 3227.9 538.0 38.428s 51.224,605.F No interactionError 24 336.7 14.0Total 35 4231.0The first and the third null hypotheses are rejected. a) Since 38.428 is larger than 2.51, we reject the hypothesis that there is no interaction and say that there is significant interaction.b) Drivers 2 and 4 are in the columns. There are 4R rows, 3C columns and 3P measurements per cell. Of course ,24234)1( PRC the number of degrees of freedom for 'within' or 'error.' From the outline, we have for Bonferroni confidence intervals for column means PRMSWtxxPRCm2121212. This becomes, for ,1m 333.2064.2583.0120.142064.2250.31667.3022432322PRMSWtxx15.358.0 This indicates no significant difference.4/18/02 252y0232For Scheffe intervals for column means use PRMSWFCxxPRCC211,12121. So 32 98.3583.333.240.32583.121422)250.31667.30(24,205. F. This indicates no significant difference.c. In your income and education regression, (i) Explain what coefficients are significant and why? (2) (ii) What income would you predict for someone with 3 years of education? (1)(iii) Make a confidence interval for the income of someone with 3 years of education using some of the information generated by Minitab below. (2) Descriptive StatisticsVariable N Mean Median TrMean StDev SEMeanEduc 32 12.000 12.000 12.071 4.363 0.771Variable Min Max Q1 Q3Educ 4.000 20.000 8.000 16.000Column Sum of Squares Sum of squares (uncorrected) of Educ = 5198.0Solution: The relevant output is:Regression AnalysisThe regression equation isIncome = 5078 + 732 Educ2Predictor Coef Stdev t-ratio pConstant 5078 1498 3.39 0.002Educ 732.4 117.5 6.23 0.000s = 2855 R-sq = 56.4% R-sq(adj) = 55.0%i) So we can state that, since the p-values are both below .05, that both coefficients are significant at the 5% level. ii) The regression can be written as EducIncome 7325078 or EducIncome 4.7325078 . So 7274)3(7325078 Incomeor 2.7275)3(4.7325078 Income.iii) From the outline The Confidence Interval is YYstYˆ0ˆ0 , where 222022ˆ1XnXXXnsseY 6.1373758590813218151025123251981233212855222 and07.11726.1373758ˆYs. If we use 042.230025.22ttn, we get 2393727407.11172042.272740Y.Please note the following from the 252 home page:The rule on p-value: If the p-value is less than the significance level (alpha) reject the null hypothesis; if the p-value is greater than or equal to the significance level, do not reject the null hypothesis.SignificanceThis is a topic that was covered under hypothesis tests. Probably the first reference I made to this was even earlier when I said that a parameter is significant if it is not zero. I later said that a nullhypothesis often says that a parameter or a difference between parameters is insignificant. If a result is significant we reject the null hypothesis.To put this more generally, a result is (statistically) significant if it is larger or smaller than would be expected by chance alone. Thus in the case of a regression coefficient the measure of significance could be the p-value, which tells us the probability of getting our actual result or something more extreme if we assume that the population value of the coefficient is zero. If the p-value is small (below our significance level), then it is unlikely that our assumption about the coefficient is correct and we say that the coefficient is significant (or significantly different from zero). Of course, the various hypothesis tests that we have discussed here are also often ways of proving significance.34/18/02 252y0232II. Do at least 4 of the following 5 Problems (at least 10 each) (or do sections adding to at least 40 points - Anything extra you do helps, and grades wrap around) . Show your work!
View Full Document
Unlocking...