3 25 99 252y9922 ECO252 QBA2 SECOND HOUR EXAM March 23 1999 Name Hour of Class Registered Circle MWF TR 10 12 12 30 2 00 night I 14 points Do all the following Use diagrams Show your work x N 5 9 Why do too many of you still believe that a probability can be negative 16 5 5 5 z P 0 z 1 22 1 P 5 x 16 P 9 9 Diagrams 3888 2 3 4 5 6 3 5 0 5 P 0 x 3 P z P 0 56 z 0 22 9 9 P 0 56 z 0 P 0 22 z 0 2123 0871 1252 0 5 2 5 P 2 x 0 P z P 0 78 z 0 56 9 9 P 0 78 z 0 P 0 56 z 0 2823 2123 0700 3 5 P x 3 P z P z 0 22 9 P 0 22 z 0 P z 0 0871 5000 5871 4 5 F 4 The Cumulative probability P x 4 P z 9 P z 0 1 1 P z 0 P 0 11 z 0 5000 0438 4562 A symmetrical interval about the mean with 64 probability We want two points x 18 and x 82 so that P x 82 x x 18 6400 From the diagram if we replace x by z P 0 z z 18 3200 The closest we can come is P 0 z 0 92 3212 So z 18 0 92 0 91 is acceptable and something in between even better and x z 18 5 0 92 9 5 8 28 or 3 28 to 13 28 13 28 5 3 28 5 z P 0 92 z 0 92 check P 3 28 x 13 28 P 9 9 2 3212 6424 7 x 055 We want a point x 055 so that P x x 055 055 From the diagram if we replace x by z P 0 z z 055 4450 The closest we can come is P 0 z 1 60 4452 So z 055 1 60 and x z 055 5 1 60 9 5 14 4 or 19 4 19 4 5 P z 1 60 check P x 19 4 P z 9 P z 0 P 0 z 1 60 5000 4452 0548 055 3 25 99 252y9922 II 6 points 2 point penalty for not trying part a Show your work A new software package has been designed to help system analysts design information systems We wish to see if there is a significant difference between the time required to develop a system using the new technology and using the current technology Two independent samples are taken The results are shown below Assume that the data represent independent samples taken from populations with the normal distribution x2 x1 Current Technology New Software 300 286 280 232 344 310 385 338 372 200 360 302 288 I will be happy to nominate everyone who had trouble dealing with n1 n 2 for the Bourbon prize for inflexibility Samples only need to be the same size when data is paired and there is no reason to believe that this data is paired a Compute s 2 the standard deviation for time required to design a system using the new software 3 Note that x1 332 714 s1 42 812 Do a two sided confidence interval for the difference between the means for the two technologies 01 2 Indicate the following What assumptions did you make about the variances of the populations from which the samples were taken Is there a significant difference between the means You must tell why 1 Solution Item x2 x 22 1 286 81796 x 2 1668 x2 278 000 2 232 53824 n2 6 3 310 96100 x 22 n 2 x 22 477168 6 278 000 2 2 4 338 114244 s2 n2 1 5 5 200 40000 2692 800 s 2 51 892 6 302 91204 Total 1668 477168 b Solution Assume equal variances From Table 3 of the Syllabus Supplement Interval for Confidence Hypotheses Test Ratio Critical Value Interval Difference H 0 0 d 0 d cv 0 t 2 sd d t 2 sd t between Two H 1 0 sd Means 1 1 sd s p 1 2 unknown n1 n 2 n1 1 s12 n2 1 s22 2 variances s p DF n1 n 2 2 n1 n2 1 assumed equal b x1 332 714 s12 42 812 2 1832 867 x 2 278 000 s 22 2692 800 d x1 x 2 54 714 01 DF n1 n 2 2 7 6 2 11 t 11 005 3 106 3 25 99 252y9922 2 n 1 s12 n 2 1 s 22 6 1832 867 5 2692 800 10997 202 13464 000 s p2 1 2223 746 n1 n 2 2 11 11 s d s p 1 1 1 1 1 1 223 746 688 3022 26 236 s p2 n1 n 2 7 6 n1 n 2 d t s d 54 714 3 106 26 236 54 7 81 5 or 16 8 to 136 2 This interval 2 includes 0 Since the interval includes zero we can say that there is no significant difference between the means Or if the null hypothesis is H 0 0 or H 0 1 2 0 or H 0 1 2 we cannot reject it b Alternate Solution Note this method was not covered in Spring 2000 If you have studied this material and want an extra credit question on it please tell me in advance Assume unequal variances From Table 3 of the Syllabus Supplement Interval for Confidence Hypotheses Test Ratio Critical Value Interval Difference between Two Means unknown variances assumed unequal H 0 0 H 1 0 s12 s22 1 2 sd n1 n2 Same as H2 s12 s22 0 1 2 H 1 1 2 n n if 0 2 1 0 d t 2 sd DF s12 2 n1 s12 1832 867 261 8381 n1 7 s 22 2671 800 448 800 n2 6 s12 s 22 710 6381 n1 n 2 DF 2 d 0 sd d cv 0 t 2 sd n1 1 s12 s 22 n1 n 2 t s 22 2 n2 n2 1 sd s12 s 22 710 6381 26 656 n1 n 2 2 2 s12 s 22 n1 n2 n1 1 n2 1 710 6381 2 261 8381 2 448 800 2 6 9 77 so use 9 degrees of freedom 5 t 9005 3 250 so d t s d 54 714 3 250 26 656 54 7 86 6 or 31 9 to 141 4 2 Conclusion is the same as for a 3 3 25 99 252y9922 III Do at least 3 of the following 5 Problems at least 10 each or do sections adding to at least 30 points Anything extra you do helps and grades wrap around Show your work State H 0 and H1 where applicable 1 A firm is currently contracting with firm 1 for delivery of raw …
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