252y0111 2 28 01 ECO252 QBA2 FIRST HOUR EXAM February 20 2001 Name key Hour of class registered Class attended if different Show your work Make Diagrams I 14 points Do all the following x N 3 9 9 3 9 3 z P 1 33 z 0 67 1 P 9 x 9 P 9 9 P 1 33 z 0 P 0 z 0 67 4082 2486 6568 9 3 0 3 z P 0 33 z 0 67 2 P 0 x 9 P 9 9 P 0 33 z 0 P 0 z 0 67 1293 2486 3779 28 3 9 3 z P 0 67 z 3 67 3 P 9 x 36 P 9 9 P 0 z 3 67 P 0 z 0 67 4999 2486 2513 5 3 P z 0 22 4 P x 5 P z 9 P z 0 P 0 z 0 22 5 0871 5871 0 3 5 3 z P 0 89 z 0 33 5 P 5 x 0 P 9 9 P 0 89 z 0 P 0 33 z 0 3133 1293 1840 6 A symmetrical interval about the mean with 87 probability We want two points x 065 and x 935 so that P x 065 x x 935 8700 Make a diagram If you do a diagram for z it will show two points z 065 and z 935 z 935 which has 93 5 above it is below zero and z 065 is above zero Since zero is halfway between these two points the diagram will show 87 split between the two sides of zero so that 43 5 is between z 935 and zero and 43 5 is between zero and z 065 The probability below z 935 and the probability above z 065 are both 6 5 From the diagram if we replace x by z P 0 z z 065 4350 The closest we can come is P 0 z 1 51 4345 So z 065 1 51 and x z 065 3 1 51 9 3 13 59 or 10 59 to 16 59 To check this note that P 10 59 x 16 59 16 59 3 10 59 3 P z P 1 51 z 1 51 9 9 2 P 0 z 1 51 2 4345 8690 87 We want a point x 125 so that P x x 125 125 Make a diagram The diagram for z will show one point z 125 which has 12 5 above it and 87 5 below it and is above zero because zero has only 50 below it Since zero has 50 above it the diagram will show 37 5 between zero and z 125 7 x 125 From the diagram if we replace x by z P 0 z z 125 3750 The closest we can come is P 0 z 1 15 3749 So z 125 1 15 and x z 135 3 1 15 9 3 10 35 or 13 35 13 35 3 P z 1 15 To check this note that P x 13 35 P z 9 P z 0 P 0 z 1 15 5 3749 1251 12 5 2 252y0111 2 21 01 II 6 points 2 point penalty for not trying part a The Watched Potts Bank of Pottstown wishes to monitor the debt to equity ratio of the firms to which it has made commercial loans A random sample of 8 of the firms is taken Assume that the bank is sampling from a population with a normal distribution firm 1 2 3 4 5 6 7 8 Debt equity ratio 1 31 1 05 1 45 1 21 1 19 1 78 1 37 1 41 a Compute the sample standard deviation 3 s of the debt to equity ratios Show your work b Compute a 99 confidence interval for the mean debt to equity ratio 3 Solution a x Ratio Firm x2 x 10 77 1 7161 x1 1 31 1 34625 n 8 2 1 05 1 1025 3 1 45 2 1025 2 2 4 1 21 x n x 14 8347 1 4641 8 1 34625 2 s2 5 1 19 1 4161 n 1 1 78 7 6 3 1684 s 0 2190 70 047941 or1 37 1 8769 8 1 41 1 9881 Total 10 77 14 8347 b From the problem statement 01 From Table 3 of the syllabus supplement if the population variance is unknown sx s x t 2 s x and t n 1 t 7005 3 499 2 0 047941 0 2190 0 07741 So 8 8 n 1 34625 3 499 0 07741 1 346 0 271 or 1 075 to 1 617 3 252y0111 2 21 01 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 Show your work State H 0 and H 1 where appropriate You have not done a hypothesis test unless you have stated your hypotheses run the numbers and stated your conclusion Use a 95 confidence level unless another level is specified Error On this page 1 42 should read 1 32 Remove section f 1 The Watched Potts Bank of Pottstown is a subsidiary of the Umongous Bancorp The Bancorp is threatening to remove the current officers of the bank if the firms to which they are making loans have an average debt to equity ratio that exceeds 1 32 Are the officers in trouble The data are on the previous page For your convenience the data are repeated below firm 1 2 3 4 5 6 7 8 Debt equity ratio 1 31 1 05 1 45 1 21 1 19 1 78 1 37 1 41 Test to see if the mean is above 1 32 using the sample mean and standard deviation you found in part II a Choose the correct null hypothesis from the list below and write the alternative hypothesis 2 H 0 x 1 32 H 0 x 1 32 H 0 p 1 32 H 0 p 1 32 H0 H0 H0 H0 H0 H0 x H0 x H0 x 1 32 1 32 1 32 1 32 1 32 1 32 1 32 1 32 b Find a critical value appropriate for this problem using a confidence level of 95 3 c Use your critical value to test the hypothesis State clearly whether you reject the null hypothesis 2 d Repeat the test using i a test ratio 2 and ii a confidence interval 2 e Test the hypothesis that the average debt to equity ratio is exactly 1 32 at the 95 confidence level 2 Solution From Table 3 of the Syllabus Supplement It s remarkable how many of you don t know that the mean is above 1 32 means 1 32 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 …
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