252y0561s 11/14/05ECO252 QBA2 SECOND EXAMNov 8-9, 2005TAKE HOME SECTION- Name: ________Key_________________Note that complete solutions for 1) and 2) were only worked out for two versions of these problems. III. Neatness Counts! Show your work! Always state your hypotheses and conclusions clearly. (19+ points)1) A state is trying to figure out whether the background on highway signs makes a difference. In order to do this two samples of 15 individuals are shown a number of slides rapidly. The slides have either a green or a red background. You are trying to find out whether there is a difference between the number of slides correctly read between those with a red or a green background. To do so you will compare the mean or median as appropriate to the distribution. To personalize the data, look at the third digit from the end to decide what red data you will use. Call the column that you pick rj and compute a column called dj with the formula dj = green – rj. (Example: Seymour Butz’s student number is 976512, so he picks column 5 and used d5 = green – r5.) Tell me what column you are using! If you compare means state your hypotheses both in terms of 1 and 2 and in terms of 21D. Row green r0 r1 r2 r3 r4 r5 r6 r7 r8 r9 1 8 9 5 7 6 5 8 9 7 10 5 2 10 10 9 10 12 6 9 7 9 9 6 3 6 9 5 11 7 5 6 6 10 7 6 4 5 4 7 9 10 6 10 7 11 10 7 5 9 9 6 8 12 5 7 8 11 6 10 6 7 9 7 8 10 11 7 7 6 6 10 7 3 8 9 8 9 8 5 6 11 7 10 8 7 7 8 7 11 8 6 6 6 8 13 9 7 5 7 9 9 7 11 7 9 8 9 10 3 6 5 9 8 9 9 7 7 7 6 11 6 10 7 7 7 10 10 5 7 9 8 12 6 8 5 7 6 7 7 7 7 11 6 13 8 8 8 10 11 7 7 10 9 6 7 14 3 9 7 7 13 6 8 7 8 8 8 15 10 9 7 6 8 5 8 6 8 7 10Minitab computed some basic statistics from the data which will help you in some parts of this problem.Variable N Mean SE Mean StDev Minimum Q1 Median Q3 Maximumgreen 15 6.533 0.601 2.326 3.000 5.000 7.000 8.000 10.000r0 15 8.000 0.458 1.773 4.000 7.000 9.000 9.000 10.000r1 15 6.800 0.355 1.373 5.000 5.000 7.000 8.000 9.000r2 15 8.200 0.368 1.424 6.000 7.000 8.000 9.000 11.000r3 15 9.267 0.581 2.251 6.000 7.000 9.000 11.000 13.000r4 15 7.000 0.488 1.890 5.000 5.000 7.000 8.000 11.000r5 15 7.867 0.435 1.685 5.000 7.000 8.000 9.000 11.000r6 15 7.000 0.324 1.254 5.000 6.000 7.000 7.000 10.000r7 15 8.400 0.456 1.765 6.000 7.000 8.000 10.000 11.000r8 15 7.933 0.408 1.580 6.000 7.000 8.000 9.000 11.000r9 15 8.067 0.573 2.219 5.000 6.000 8.000 10.000 13.000a) Display the numbers that you are using in columns and compute a sample mean and sample standard deviation for the d column. (1)In this problem assume that the red and green data are two independent samples. Use a confidence level of95%.b) Assume that you believe that the normal distribution does not apply to the data and compare the means or medians as appropriate. (4)c) You suspect that the data has the Normal distribution. Test to see if the Normal distribution applies. Usea test that I taught you. (3)1252y0561s 11/14/05d) You decide that the Normal distribution applies to the data, but do not know if the variances are equal.Test them for equality. (1)e) You conclude that the underlying distributions are Normal and that the population variances are equal.Compare the means or medians as appropriate. Use a test ratio, critical value or a confidence interval (4) or all three (6). [15]f) (Extra credit) You conclude that the underlying distributions are Normal and that the population variances are not equal. Compare the means or medians as appropriate. Use a test ratio, critical value or a confidence interval (5) or all three (7)2) In fact the data on the previous page applies to a single sample of 15 individuals. That is the first line of your worksheet tells you how the first person in the sample did when showed the same slides with red or green backgrounds. This applies to a) and b) in this question. Use a confidence level of 95%.a) Assume that you believe that the normal distribution does not apply to the data and compare the means or medians as appropriate. (3)b) You assume that the data has the Normal distribution. Compare the means or medians as appropriate. (3)c) For any part of one of these problems (tell me which one!), compute a confidence interval that you would use to compare means if your alternate hypothesis was .:121H(2)[23]d) For the same part as you used in c), find a p-value for the null hypothesis. (2) [25]These results are all supposed to look to me as you did them by hand. But what I don’t know won’t hurt me. If you want to check your results by computer, you might try to use the following Minitab routine. If you put green in C1 and label columns with headings like rj, dj and dsqj (Seymour called his green, r5, d5and dsq5.) The routine below with appropriate changes to rj, dj and dsqj, will compute much of the stuff above, though not in the right order. Note that to do a Wilcoxon signed rank test by hand, you will have todrop all zeroes from the d column.Computations for comparing c1 and rjMTB > let dj = c1 – rjMTB > let dsqj = rj *rjMTB > print c1 rj dj dsqjMTB > describe c1 rj djMTB > sum djMTB > ssq djMTB > TwoSample c1 rj;SUBC> Pooled.MTB > TwoSample c1 rj.MTB > Paired c1 'rj'.MTB > VarTest c1 'rj';SUBC> Unstacked.MTB > WTest 0.0 'dj';SUBC> Alternative 0.MTB > Mann-Whitney 95.0 c1 'rj';SUBC> Alternative 0.If you want to fake the calculations for the Mann-Whitney test, try this.Procedure for setting up Mann-Whitney Test#c1 is green, c2 is rj, c3 is difference.# Mann Whitney TestMTB > Stack c1 c2 c5;SUBC> Subscripts c6.MTB > Rank c5 c7.MTB > Unstack (c7);SUBC> Subscripts c6;SUBC> After;SUBC> VarNames.MTB > sum c8MTB > sum c9MTB > print c1 c8 c2 c9 #The rest is up to you.2252y0561s 11/14/05If you want to fake the calculations for the Wilcoxon signed rank test, try this. Unfortunately, I know no good way to remove the zeros or change the signs except by hand.Procedure for setting up Wilcoxon Signed Rank Test#c1 is green, c2 is rj, c3 is difference.MTB > Let c3 = c1-c2 #Maybe you already did this.# Wilcoxon signed rank testMTB > let c10 = c3MTB > #remove zeroes from c10. (Just use delete on the
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