252y0551h 10 31 05 Open in Print Layout format ECO252 QBA2 FIRST EXAM October 17 18 2005 TAKE HOME SECTION Name Student Number and class IV Do sections adding to at least 20 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 Answers without reasons are not usually acceptable Neatness counts 1 Prem S Mann modified Exhibit T1 According to a 1992 survey 45 of the American population would support higher taxes to pay for health insurance A state government is considering offering a health insurance plan and took a survey of 400 residents and found that 50 would support higher taxes Use this result to test whether the results of the 1992 survey apply in the state Use a 99 confidence level Personalize these results as follows Change 45 by replacing 5 by the last digit of your student number We will call your result the proportion of interest Example Seymour Butz s student number is 976512 so he changes 45 to 42 42 is his proportion of interest a State the null and alternative hypotheses in each case and find a critical value for each case What is the reject region Compute a test ratio and find a p value for the hypothesis in each case 12 i The state government wants to test that the fraction of people who favored higher taxes for health insurance is below the proportion of interest ii The state government wants to test that the fraction of people who favored higher taxes for health insurance is above the proportion of interest iii The state government wants to test that the fraction of people who favored higher taxes for health insurance is equal to the proportion of interest 12 b Extra credit Find the power of the test if the true proportion is 50 and i The alternate hypothesis is that the fraction of people is above the proportion of interest 2 ii The alternate hypothesis is that the fraction of people does not equal the proportion of interest 2 c Assuming that the proportion of interest is correct is the sample size given above adequate to find the true proportion within 005 1 2 of 1 Don t say yes or no without calculating the size that you actually need 2 d If the alternate hypothesis is that the fraction of people is above the proportion of interest create an appropriate confidence interval for the hypothesis test 2 20 Solution a From the formula table we have Interval for Confidence Hypotheses Test Ratio Critical Value Interval Proportion H 0 p p0 p p0 p p z s p p z 2 sp pq n q 1 p p H1 p p0 z p cv 0 2 p p0 q0 n q 0 1 p 0 p 1 252y0551h 10 31 05 Open in Print Layout format Note that my rule on grading parts like ii below is to assume that your alternate hypothesis was correct and to ask if the critical values agree with it p0 q0 40 60 0006 024495 n 400 i The state government wants to test that the fraction of people who favored higher taxes for health insurance is below the proportion of interest H 0 p 40 H 1 p 40 n 400 p 50 and 01 z 01 2 327 The critical value must be below 40 p cv p 0 z p 40 2 327 024495 3430 The reject Version 0 a p 0 40 p 50 40 P z 4 08 1 zone is below 3430 pval P p 50 P z 024495 ii The state government wants to test that the fraction of people who favored higher taxes for health insurance is above the proportion of interest H 0 p 40 H 1 p 40 n 400 p 50 and 01 z 01 2 327 The critical value must be above 40 p cv p 0 z p 40 2 327 024495 4570 The 50 40 P z 4 08 0 reject zone is above 4570 pval P p 50 P z 024495 iii The state government wants to test that the fraction of people who favored higher taxes for health insurance is equal to the proportion of interest 12 H 0 p 40 H 1 p 40 n 400 p 50 and 01 z 005 2 576 The critical value must be on either side of 40 p cv p 0 z 2 p 40 2 576 024495 40 0631 The reject zone is below 3369 and 50 40 2 P z 4 08 0 above 4631 pval 2 P p 50 2 P z 024495 b i H 1 p 40 P p 4570 p1 50 p pq 5 5 5 025 n 400 20 4570 5 P z P z 1 72 5 4573 0427 Power 1 0427 9573 025 ii H 1 p 40 P 3369 p 4631 p1 50 4631 5 3369 5 P z 025 025 P 6 52 z 1 48 5 4306 0694 Power 1 0694 9306 pqz 2 40 60 2 576 2 c p 0 40 n 63703 42 This is above 400 so the sample size is e2 005 2 inadequate pq 5 5 5 025 p p z s p 5 2 327 025 4418 If the null n 400 20 hypothesis is H 0 p 40 we must reject it d s p 2 252y0551h 10 31 05 Open in Print Layout format Note that my rule on grading parts like ii below is to assume that your alternate hypothesis was correct and to ask if the critical values agree with it p0 q0 41 59 0006048 0245917 n 400 i The state government wants to test that the fraction of people who favored higher taxes for health insurance is below the proportion of interest H 0 p 41 H 1 p 41 n 400 p 50 and 01 z 01 2 327 The critical value must be below 40 p cv p 0 z p 41 2 327 0245917 3528 The Version 1 a p 0 41 p reject zone is below 3528 50 41 pval P p 50 P z P z 3 65 5 4999 9999 0245917 ii The state government wants to test that the fraction of people who favored higher taxes for health insurance is above the proportion of interest H 0 p 41 H 1 p 41 n 400 p 50 and 01 z 01 2 327 The critical value must be above 41 p cv p 0 z p 41 2 327 0245917 4672 The reject zone is above 467270 50 41 pval P p 50 P z P z 3 66 5 4999 0001 0245917 iii The state government wants to test that the fraction of people who favored higher taxes for health insurance is equal to the proportion of interest 12 H 0 p 41 H 1 p 41 …
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