2 17 99 252y9911 ECO252 QBA2 FIRST HOUR EXAM February 23 1999 Name Hour of Class Registered Circle MWF TR 10 12 12 30 2 00 Show your work Make Diagrams I 14 points Do all the following x N 7 6 3 7 1 7 z P 1 33 z 0 67 1 P 1 x 3 P 6 6 P 1 33 z 0 P 0 67 z 0 4082 2486 1596 11 7 11 7 z P 3 00 z 0 67 2 P 11 x 11 P 6 6 P 3 00 z 0 P 0 z 0 67 49865 2486 74725 9 7 14 7 z P 3 50 z 0 33 3 P 14 x 9 P 6 6 P 3 50 z 0 P 0 z 0 33 4998 1293 6291 For 3 5 see SS Table 18 0 7 P z 1 17 4 P x 0 P z 6 P 1 17 z 0 P z 0 3790 5000 8790 12 5 7 9 7 z P 0 33 z 0 92 5 P 9 x 12 5 P 6 6 P 0 z 0 92 P 0 z 0 33 3212 1293 1919 6 A symmetrical interval about the mean with 94 probability We want two points x 97 and x 03 so that P x 97 x x 03 9400 From the diagram if we replace x by z P 0 z z 03 4700 The closest we can come is P 0 z 1 88 4699 So z 03 1 88 and x z 03 7 1 88 6 7 11 28 or 4 28 to 18 28 7 x 11 We want a point x 11 so that P x x 11 11 From the diagram if we replace x by z P 0 z z 11 3900 The closest we can come is P 0 z 1 23 3907 So z 11 1 23 and x z 11 7 1 23 6 7 7 38 or 14 38 2 17 99 252y9911 II 6 points 2 point penalty for not trying part a A sample of the number of days it took a broker to sell houses appears below Assume that we were sampling from a normal distribution Home 1 2 3 4 5 a Compute the sample variance Days 48 63 88 101 31 s of the number of days it takes to sell a house Show your work 3 b Compute a 90 confidence interval for the mean time that it takes to sell a house 3 Solution a x Days Home x2 x 331 2304 x 1 48 66 2 n 563 2 3969 3 88 7744 2 2 4 101 10201 x n x 25179 5 66 2 2 s2 5 31 961 n 1 331 425179 Total 816 7 or s 28 5780 b From the problem statement 10 From Table 3 of the syllabus supplement if the population variance is unknown s x t 2 s x and t n 1 t 405 2 132 2 816 7 28 5780 12 7805 So 66 2 2 132 12 7805 66 2 27 25 5 n 5 or 38 95 to 93 45 sx 2 2 17 99 252y9911 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 H 1 where appropriate 1 The broker claims that the average time to sell a house is at most 50 days a State the Hypotheses that you are testing 2 b Using the data from the previous problem page 2 test the hypotheses 95 confidence level using i A test ratio 2 ii Critical values 2 iii A confidence interval 2 c Find an approximate p value for the null hypothesis 1 d Now find a 90 confidence interval for the standard deviation 3 Solution From Table 3 of the Syllabus Supplement Interval for Confidence Hypotheses Interval Mean H 0 0 x z 2 x known Test Ratio Mean unknown a x t 2 s x DF n 1 H 0 50 x 0 x x cv 0 z 2 x x 0 sx x cv 0 t 2 s x z H 1 0 H 0 0 t H 1 0 Critical Value 4 0 50 DF n 1 4 05 t n 1 t 05 2 132 H 1 50 From the previous page x 66 2 and s x b i Test Ratio t s n 816 7 28 5780 12 7805 5 5 x 0 66 2 50 1 2676 This is in the accept region sx 12 7805 below 2 132 so do not reject H 0 ii Critical Value Since this is a one sided test xcv 0 t s x 50 2 132 12 7805 50 27 24 or 77 24 This means that we reject H 0 if the sample mean is above 77 24 since x 66 2 is below this critical value do not reject H 0 iii Confidence Interval Since this is a one sided test x t s x 66 2 2 1323 12 7805 66 2 27 24 or 38 95 This does not contradict H 0 50 because any mean in the range 38 95 to 50 satisfies both statements so do not reject H 0 4 c From the t table t 1 2676 is smaller than t 10 1 533 but is larger than t 415 1 190 so that for this 1 sided test we can say 15 pvalue 10 d From page 1 of the Syllabus Supplement n 1 s 2 2 10 Since 2 05 and pvalue P t 1 2676 2 2 n 1 s 2 2 1 2 DF n 1 4 and 1 95 look up 205 4 9 4677 2 and 295 4 0 7107 3 So n 1 s 2 205 2 n 1 s 2 295 or 4 816 7 2 4 816 7 9 4877 0 7107 or 344 32 2 4596 59 Finally taking square roots 18 555 67 80 2 17 99 252y9911 2 A new broker opens in town and a sample of nine houses gives the following days to sale Home Days 1 62 2 28 3 114 4 113 5 71 6 29 7 53 8 54 9 34 Assuming that the distribution is not normal a Find a confidence level for the following interval for the median 29 113 3 b Test the hypotheses that the median is at most 50 at the 10 level 5 c If we have a sample of 300 numbers in order and take the 15 th from each end what would the confidence level be 3 d Extra credit What numbers would we use if we wanted a 99 confidence interval for the median and had a sample of 300 numbers 3 Solution a If we put the numbers in order we get 28 29 34 53 54 62 71 113 114 Thus we want …
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