252y0221 3 29 02 ECO252 QBA2 Name KEY SECOND HOUR EXAM Hour of Class Registered Circle Open this document in Page Layout March 19 2002 MWF 10 11 TR 12 30 2 00 Hour of Class Attended If Different I 14 points Do all the following Diagrams will help x N 5 9 Probabilities still can t be negative 16 5 3 5 P z P 0 89 z 1 22 9 9 P 0 89 z 0 P 0 z 1 22 3133 3888 7021 1 P 3 x 16 5 5 0 5 z P 0 56 z 0 2123 2 P 0 x 5 P 9 9 14 5 25 5 z P 3 33 z 1 00 3 P 25 x 14 P 9 9 P 3 33 z 0 P 0 z 1 00 4996 3413 8409 4 5 2 5 z P 0 78 z 0 11 4 P 2 x 4 P 9 9 P 0 78 z 0 P 0 11 z 0 2823 0438 2385 2 5 5 F 2 The Cumulative probability up to 2 P x 2 P z 9 P z 0 33 P z 0 P 0 33 z 0 5 1293 3707 6 A symmetrical interval about the mean with 37 probability 6 We want two points x 698 and x 315 so that P x 685 x x 315 3700 Make a diagram Showing 5 in the middle at the center of a 37 region split into two areas with probabilities of 18 5 From the diagram if we replace x by z P 0 z z 315 1850 The closest we can come is P 0 z 0 48 1844 or P 0 z 0 49 1879 Since neither of these is much closer than the other use z 315 0 485 and x z 315 5 0 485 9 5 4 365 or 0 635 9 365 5 0 635 5 z to 9 365 To check this note that P 0 635 x 9 365 P 9 9 1844 1879 P 0 485 z 0 485 2 P 0 z 0 485 2 3723 37 2 Of course z 315 0 48 or z 315 0 49 is perfectly acceptable 7 x 02 We want a point x 02 so that P x x 02 02 Make a diagram of z showing zero in the middle 4800 between 0 and z 02 and 02 above z 02 From the diagram if we replace x by z P 0 z z 02 4800 The Normal table says P 0 z 2 05 4798 which is the closest we can come to 4800 So z 02 2 05 and x z 02 5 2 05 9 23 45 To check this note 23 45 5 P x 23 45 P z P z 2 05 9 P z 0 P 0 z 2 05 5 4798 0202 02 2 252y0221 3 29 02 II 6 points 2 point penalty for not trying part a Show your work Do not answer part b with a yes or no unless you have stated your hypotheses a According to your text a study was made to compare ages of purchasers of Crest with nonpurchasesrs yielding the following results These are two independent samples taken from an approximately normal population Row crest nocrest x1 1 2 3 4 5 6 7 8 x2 34 35 23 44 52 46 28 48 28 22 44 33 55 63 45 31 The Minitab describe function gave the following results for the nocrest column Variable N Mean Median StDev nocrest 8 40 12 38 50 14 11 a Compute the standard deviation s1 for the crest column Show your work 3 b Compute a 90 confidence interval for the difference betweeen the two population means 1 and 2 on the assumption that these are independent samples taken from approximately normal populations with similar variances According to your confidence interval is there a significant difference between the population means Why 3 Solution a n1 8 310 12754 Row x x2 1 1 2 3 4 5 6 7 8 34 35 23 44 52 46 28 48 1 1156 1225 529 1936 2704 2116 784 2304 b From Table 3 of the Syllabus Supplement Interval for Confidence Hypotheses Interval Difference H 0 0 d t 2 sd between Two H 1 0 Means 1 1 sd s p 1 2 unknown n1 n 2 variances DF n1 n 2 2 assumed equal x1 s12 x 1 n1 x 2 1 310 38 75 8 n1x12 n1 1 12754 8 38 75 2 741 5 105 9285 7 7 s1 10 29161 Test Ratio t d 0 sd s 2p Critical Value d cv 0 t 2 sd n1 1 s12 n2 1 s22 n1 n 2 2 3 252y0221 3 29 02 x1 38 75 s12 105 92857 s1 10 29161 x2 40 12 s2 14 11 s22 199 0921 d x1 x2 38 75 40 12 1 37 DF n1 n2 2 8 8 2 14 10 s 2p n1 1 s12 n2 1 s22 n1 n2 2 7 105 92857 7 199 0921 105 92857 199 0921 14 1 761 152 510335 t 05 14 2 sd s p 1 1 n1 n 2 152 510335 1 1 8 8 152 510335 25 38 12758375 6 17475 Confidence Interval d t 2 sd 1 37 1 761 6 17475 1 37 10 87 or 12 24 to 9 50 The interval includes 0 so there is no significant difference between the means Formally our hypotheses H 0 0 H 0 1 2 H 0 1 2 0 are H 0 or or We do not reject H 1 H1 1 2 H1 1 2 0 1 2 0 4 252y0221 3 29 02 III Do at least 3 of the following 4 Problems at least 10 each or do sections adding to at least 30 points Anything extra you do helps and grades wrap around You must do problem 2a Show your work State H 0 and H1 where applicable Do not answer a question yes or no without citing a statistical test Use a 95 confidence level unless another level is specified 1 For your convenience data is repeated from the previous page Use a 90 confidence level in this problem Row crest nocrest x1 1 2 3 4 5 6 7 8 x2 34 35 23 44 52 46 28 48 Variable nocrest 28 22 44 33 55 63 45 31 N 8 Mean 40 12 Median 38 50 StDev 14 11 a Test the hypothesis that the mean age of Crest buyers is lower than the mean for those who did not buy Crest Assume that these are independent samples taken from approximately normal populations with similar variances and i State your null and alternate hypotheses 2 ii Find a critical value for the difference between the sample means and use it to test your hypothesis 2 iii Repeat the test using a test ratio and find an approximate p value for the hypothesis 3 iv Repeat …
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