i. Test Ratio: or for large samplesii. Critical Value: or for large samples (from table 3) .iii. Confidence Interval: or for large samples252y0551h 10/31/05 (Open in ‘Print Layout’ format) ECO252 QBA2 FIRST EXAMOctober 17-18 2005TAKE 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) . Showyour work! State 0H and 1H 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)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 IntervalHypotheses Test Ratio Critical ValueProportionpszpp2nqpsppq 10100:H:Hpppppppz0pcvzpp20nqpp00001 pq 1Exhibit T1: According to a 1992 survey, 45% of the American population would support higher taxes to payfor 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.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.Version 0 a) 40.0p 024495.0006.40060.40.00nqpp(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,40.:0pH ,40.:1pH,400n50.p and 01.. 327.201.z. The critical value must be below .40. 024495.327.240.0pcvzpp= .3430. The ‘reject’ zone is below .3430. 108.4024495.40.50.50. zPzPpPpval(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.,40.:0pH ,40.:1pH,400n50.p and 01.. 327.201.z.The critical value must be above .40. 024495.327.240.0pcvzpp = .4570 . The ‘reject’ zone is above .4570. 008.4024495.40.50.50. zPzPpPpval(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],40.:0pH ,40.:1pH,400n50.p and 01.. 576.2005.z.The critical value must be on either side of .40. 0631.40.024495.576.240.20pcvzpp . The ‘reject’ zone is below .3369 and above .4631. 008.42024495.40.50.250.2 zPzPpPpvalb) (i) ,40.:1pH 50.4570.1 ppP .025.205.4005.5.npqp 0427.4573.5.72.1025.5.4570. zPzP9573.0427.1 Power (ii) ,40.:1pH 50.4631.3369.1 ppP025.5.4631.025.5.3369.zP 0694.4306.5.48.152.6 zP9306.0694.1 Powerc) 40.0p 42.63703005.576.260.40.2222epqzn This is above 400, so the sample size is inadequate. d) .025.205.4005.5.nqpsp 4418.025.327.25. pszpp If the null hypothesis is ,40.:0pHwe must reject it.2252y0551h 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.Version 1 a) 41.0p 0245917.0006048.40059.41.00nqpp(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,41.:0pH ,41.:1pH,400n50.p and 01.. 327.201.z. The critical value must be below .40. 0245917.327.241.0pcvzpp= .3528. The ‘reject’ zone is below .3528. 9999.4999.5.65.30245917.41.50.50. zPzPpPpval(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.,41.:0pH ,41.:1pH,400n50.p and 01.. 327.201.z.The critical value must be above .41. 0245917.327.241.0pcvzpp = .4672 . The ‘reject’ zone is above .467270.
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