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252x0551 10/11/05 (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. (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. (5) (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. (4) (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. (3) [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) [14] 1 252x0551 10/11/05 (Open in ‘Print Layout’ format) 2. Customers at a bank complain about long lines and a survey shows a median waiting time of 8 minutes. Exhibit T2: Changes are made and a new survey of 32 customers is taken. The times are as follows: 6.7 7.6 8.5 6.3 7.6 8.9 5.7 7.7 9.0 5.5 7.7 9.1 4.9 8.0 9.3 7.0 8.0 9.5 7.1 8.1 9.6 7.2 8.2 9.7 7.3 8.4 9.8 7.3 8.4 10.3 7.4 8.4 Personalize the data as follows: Use the second-to-last digit of your student number – subtract it from the last digit of every single number. (Example: Seymour Butz’s student number is 976512, so he changes {6.7 6.3 5.7 ….} to {6.6 6.1 5.6 …}) You may drop any number in the data set that you use for this problem that is exactly equal to 8. Use .05 . Do not assume that the population is Normal! a. Test that the median is below 8. State your null and alternate hypotheses clearly. (2) [16] b. (Extra credit) Find a 95% (or slightly more) two-sided confidence interval for the median. (2) 3. Use the personalized data from Problem 2 (but do not drop any numbers.) and test that the mean is below eight. Minitab found the following the original numbers, which were in C4: Descriptive Statistics: C4 Variable N Mean SE Mean StDev Minimum C4 32 7.944 0.228 1.291 4.900 Sum of squares (uncorrected) of C4 = 2070.94 Q1 7.225 Median 8.000 Q3 8.975 Maximum 10.300 I do not know the values for your numbers, but the following (copied from last year’s exam) should be useful: x a x na, x a2 x 2 2a x na2 . Your value of a is negative or zero. Can you say that the population mean is below 8? Use .05 . a. State your null and alternative hypotheses (1) b. Find a critical value for the sample mean and a ‘reject’ zone. Make a diagram! (1) c. Do you reject the null hypothesis? Use the diagram to show why. (1) d. Find an approximate p-value for the null hypothesis. Make sure that I know where you got your results. e. Test to see if the population standard deviation is 2. (2) f. (Extra credit) What would the critical values be for this test of the standard deviation? (1) g. Assume that the population standard deviation is 2, restate your critical value for the mean and create a power curve for your test. (5) h. Assume that the population standard deviation is 2 and find a 94% (this does not say 95%!) two sided confidence interval for the mean. (1) i. Using a 96% confidence level and assuming that the population standard deviation is 2, test that the mean is below 8. (1) j. Using a 96% confidence interval an assuming that the population standard deviation is 2, how large a sample do you need to have an error in the mean of .005 ? (1) [30] 2