3 22 99 252y9921 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 7 9 Why do too many of you still believe that a probability can be negative 16 7 7 7 z P 0 z 1 00 1 P 7 x 16 P 9 9 3413 2 3 4 5 6 3 7 0 7 P 0 x 3 P z P 0 78 z 0 44 9 9 P 0 78 z 0 P 0 44 z 0 2823 1700 1123 0 7 2 7 P 2 x 0 P z P 1 00 z 0 78 9 9 P 1 00 z 0 P 0 78 z 0 3413 2823 0590 3 7 P x 3 P z P z 0 44 9 P 0 44 z 0 P z 0 1700 5000 6700 4 7 F 4 The Cumulative probability P x 4 P z 9 P z 0 33 P z 0 P 0 33 z 0 5000 1293 3707 A symmetrical interval about the mean with 68 probability We want two points x 16 and x 84 so that P x 84 x x 16 6800 From the diagram if we replace x by z P 0 z z 16 3400 The closest we can come is P 0 z 0 99 3389 So z 16 0 99 1 00 is acceptable and something in between even better and x z 16 7 0 99 9 7 8 91 or 1 91 to 15 91 15 91 7 1 91 7 z P 0 99 z 0 99 check P 1 91 x 15 91 P 9 9 2 3389 6778 7 x 045 We want a point x 045 so that P x x 045 045 From the diagram if we replace x by z P 0 z z 045 4550 The closest we can come is P 0 z 1 70 4554 though 1 69 or something between the two is acceptable So z 045 1 70 and x z 045 7 1 70 9 7 15 3 or 22 3 22 3 7 P z 1 70 check P x 22 3 P z 9 P z 0 P 0 z 1 70 5000 4554 0446 2 3 22 99 252y9921 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 x1 x2 Current Technology New Software 300 276 280 222 344 310 385 338 372 200 360 302 288 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 1 276 76176 x 2 1648 x2 274 667 2 222 49284 n2 6 3 310 96100 x 22 n 2 x 22 467008 6 274 667 2 2 4 338 114244 s2 n2 1 5 5 200 40000 2671 467 s 2 53 586 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 t 2 sd d 0 d cv 0 t 2 sd t between Two H sd 1 0 1 1 Means sd s p n 1 1 s12 n2 1 s22 n1 n 2 1 2 unknown s p2 n1 n2 1 variances DF n1 n 2 2 assumed equal b x1 332 714 s12 42 812 2 1832 867 x 2 274 667 s 22 2471 467 d x1 x 2 58 047 01 DF n1 n 2 2 7 6 2 11 t 11 005 3 355 3 22 99 252y9921 3 n 1 s12 n 2 1 s 22 6 1832 867 5 2671 467 10997 202 13357 335 s p2 1 2214 049 n1 n 2 2 11 11 s d s p 1 1 1 1 1 1 2214 049 685 301 26 178 s p2 n1 n 2 7 6 n1 n 2 d t s d 58 047 3 106 26 178 58 0 87 3 or 23 3 to 139 3 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 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 467 445 2445 n2 6 s12 s 22 707 0826 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 22 707 0826 26 591 n1 n 2 2 2 s12 s 22 n1 n2 n1 1 n2 1 707 0826 2 261 8381 2 445 2445 2 6 9 79 so use 9 degrees of freedom 5 t 9005 3 250 so d t s d 58 047 3 250 26 591 58 0 86 4 or 28 4 to 144 4 2 Conclusion is the same as for a 4 3 23 99 252y9921 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 materials It will stay with firm 1 unless it can demonstrate that delivery times for firm 2 are faster The …
View Full Document
Unlocking...