2 22 99 252y9912 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 6 7 3 6 1 6 z P 1 00 z 0 43 1 P 1 x 3 P 7 7 P 1 00 z 0 P 0 43 z 0 3413 1664 1749 11 6 11 6 z P 2 43 z 0 71 2 P 11 x 11 P 7 7 P 2 43 z 0 P 0 z 0 71 4925 2611 7536 9 6 16 6 z P 3 14 z 0 43 3 P 16 x 9 P 7 7 P 3 14 z 0 P 0 z 0 43 4992 1664 6656 For 3 14 see SS Table 18 0 6 P z 0 86 P 0 86 z 0 P z 0 4 P x 0 P z 7 3051 5 8051 12 5 6 9 6 z P 0 43 z 0 93 5 P 9 x 12 5 P 7 7 P 0 z 0 93 P 0 z 0 43 3238 1664 1574 6 A symmetrical interval about the mean with 96 probability We want two points x 98 and x 02 so that P x 98 x x 02 9600 From the diagram if we replace x by z P 0 z z 02 4800 The closest we can come is P 0 z 2 05 4798 or P 0 z 2 06 4803 So z 02 2 054 though 2 05 or 2 06 are fine and x z 02 6 2 054 7 6 14 38 or 8 38 to 20 38 7 x 13 We want a point x 13 so that P x x 13 13 From the diagram if we replace x by z P 0 z z 13 3700 The closest we can come is P 0 z 1 13 3708 So z 13 1 13 and x z 11 6 1 13 7 6 7 91 or 13 91 2 22 99 252y9912 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 49 69 88 99 33 s of the number of days it takes to sell a house Show your work 3 b Compute an 80 confidence interval for the mean time that it takes to sell a house 3 Solution a x Days Home x2 x 338 2401 x 1 49 67 6 n 569 2 4761 3 88 7744 2 2 4 99 x n x 25796 5 9801 67 6 2 s2 5 33 1089 n 1 338 425796 Total s 27 1441 736 8 or b From the problem statement 20 From Table 3 of the syllabus supplement if the population variance is unknown s x t 2 s x and t n 1 t 410 1 533 2 736 8 27 1441 12 1392 So 67 6 1 533 12 1372 67 6 18 61 5 n 5 or 48 99 to 86 21 sx 2 2 22 99 252y9912 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 48 days a State the Hypotheses that you are testing 2 b Using the data from the previous problem page 2 test the hypotheses 99 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 99 confidence interval for the standard deviation 3 Solution From Table 3 of the Syllabus Supplement Interval for Confidence Hypotheses Test Ratio Critical Value Interval Mean H 0 0 x 0 x z 2 x x cv 0 z 2 x z Known H 1 0 Mean Unknown a x t 2 s x DF n 1 H 0 48 H 1 48 H 0 0 t H 1 0 x 0 sx x cv 0 t 2 s x 4 0 48 DF n 1 4 01 t n 1 t 01 3 747 From the previous page x 67 6 and s x b i Test Ratio t x s n 736 8 27 1441 12 1392 5 5 x 0 67 6 48 1 6146 This is in the accept region sx 12 1392 below 3 747 so do not reject H 0 ii Critical Value Since this is a one sided test xcv 0 t s x 48 3 747 12 1392 50 45 49 or 95 49 This means that we reject H 0 if the sample mean is above 95 49 since x 67 6 is below this critical value do not reject H 0 iii Confidence Interval Since this is a one sided test x t s x 67 6 3 747 12 1392 67 6 45 49 or 22 11 This does not contradict H 0 48 because any mean in the range 22 11 to 48 satisfies both statements so do not reject H 0 4 4 c From the t table t 1 6146 is smaller than t 05 2 132 but is larger than t 10 1 533 so that for this 1 sided test we can say 10 pvalue 05 d From page 1 of the Syllabus Supplement n 1 s 2 2 2 2 01 Since 2 005 and 1 995 look up 2 n 1 s 2 2 1 2 DF n 1 4 and 4 2005 14 8603 and 3 4 2995 0 2070 So 198 33 2 n 1 s 2 2005 14237 68 2 n 1 s 2 2995 or 4 736 8 2 Finally taking square roots 14 8603 14 083 4 736 8 0 2070 119 32 or 2 22 99 252y9912 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 34 71 3 b Test the hypotheses that the median is at most 33 at the 10 level 5 c If we have a sample of 350 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 350 numbers 3 Solution a If we put the numbers in order we get 28 …
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