4 26 02 252y0232 Page layout view ECO252 QBA2 THIRD HOUR EXAM April 18 2002 I 10 points Do all the following Name KEY Hour of Class Registered Circle MWF TR 10 12 12 30 2 00 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 1 2 3 4 ALL COLUMNS driver 1 2 3 ALL 42 000 32 000 30 667 31 333 34 000 25 000 28 000 45 000 24 667 30 667 12 667 29 333 28 333 54 667 31 250 26 556 29 778 34 667 36 889 31 972 CELL CONTENTS mpg MEAN MTB twoway mpg car driver Two way Analysis of Variance Analysis of Variance for mpg Source DF SS car 3 590 3 driver 2 76 1 Interaction 6 3227 9 Error 24 336 7 Total 35 4231 0 MS 196 8 38 0 538 0 14 0 To complete the printout divide through the MS column by MSW 14 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 H0 F 3 24 car 3 590 3 196 8 14 057s F 3 01 Car means identical driver 2 76 1 38 0 identical Interaction 6 3227 9 538 0 2 714ns 05 2 24 F 05 3 40 Driver means 6 24 2 51 No interaction 38 428s F 05 Error 24 336 7 14 0 Total 35 4231 0 The 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 R 4 rows C 3 columns and P 3 measurements per cell Of course RC P 1 4 3 2 24 the number of degrees of freedom for within or error From the outline we have for Bonferroni confidence intervals for column means RC P 1 2MSW x x t 1 2 1 2 This becomes for m 1 2m PR 2 14 0 24 2MSW 2 3 x 2 x 3 t 30 667 31 250 2 064 0 583 2 064 2 333 PR 2 12 0 58 3 15 This indicates no significant difference 4 18 02 252y0232 For Scheffe intervals for column means use 30 667 31 250 C 1 RC P 1 2MSW x x C 1 F 1 2 1 2 2 24 2 14 2 F 05 12 583 So 2 3 PR 2 3 40 2 333 583 3 98 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 Statistics Variable Educ N 32 Mean 12 000 Median 12 000 TrMean 12 071 Variable Educ Min 4 000 Max 20 000 Q1 8 000 Q3 16 000 StDev 4 363 SEMean 0 771 Column Sum of Squares Sum of squares uncorrected of Educ Solution The relevant output is 5198 0 Regression Analysis The regression equation is Income 5078 732 Educ 2 Predictor Constant Educ Coef 5078 732 4 s 2855 Stdev 1498 117 5 R sq 56 4 t ratio 3 39 6 23 p 0 002 0 000 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 Income 5078 732 Educ or Income 5078 732 4 Educ So Income 5078 732 3 7274 or Income 5078 732 4 3 7275 2 iii From the outline The Confidence Interval is Y t s where Y0 1 s Y2 s e2 n X 0 X 2 X 2 nX 2 s Y Y 1 3 12 2 2855 32 5198 32 12 2 2 0 8151025 1 81 1373758 6 and 32 590 t n 2 t 30 2 042 1373758 6 1172 07 If we use 2 025 we get Y0 7274 2 042 11172 07 7274 2393 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 pvalue is greater than or equal to the significance level do not reject the null hypothesis Significance This 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 null hypothesis 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 3 4 18 02 252y0232 II 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 State H 0 and H1 where applicable Never say yes or no without a statistical test 1 On the following pages there are printouts from two computer problems a The One way ANOVA Problem Albright Winston Zappe abbreviated An automobile parts producer has instituted an employee empowerment program in five plants Random samples of employees in each plant are asked to rate the success of the program on a 1 to 10 scale 10 being …
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