Verify-1® is quickly operational. And it authorizes returns even quickerYou set the policy, Verify-1® enforces it252 1takehome081 2/22/08 ECO252 QBA2 FIRST EXAMFebruary, 2008TAKE HOME SECTION- Name: _________________________Student Number and class time: _________________________ IV. Do at least 3 problems (at least 7 each) (or do sections adding to at least 20 points - Anything extra youdo helps, and grades wrap around) . Show your 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 reasonsusually are not acceptable. Neatness and clarity of explanation are expected. This must be turned in when you take the in-class exam. Note that answers without reasons and citation of appropriate statistical tests receive no credit. Failing to be transparent about which section of which problem you are doing can lose you credit. Many answers require a statistical test, that is, stating or implying a hypothesis and showing why it is true or false by citing a table value or a p-value. If you haven’t done it lately, take a fast look at ECO 252 - Things That You Should Never Do on a Statistics Exam (or Anywhere Else).Problem 1: (Doane and Seward) A fast food restaurant has just started serving hot cocoa. The management wishes to serve cocoa of an average temperature of 142 degrees. 24 measurements of the temperature in 10 stores are taken. You are manager of store a and will use the corresponding column, where a is the second to last digit of your student number. (For example, Seymour Butz’ student number is 543987 so he uses column x8.) If that number is zero, use column 10. You are testing to see if the mean for your store is 142. There will be a penalty if you do not make it clear what column you are using.Row x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 1 140 142 142 143 144 142 143 146 143 143 2 142 143 143 138 139 142 144 145 144 139 3 141 141 141 140 144 142 144 144 141 145 4 142 142 142 140 143 140 145 144 144 144 5 141 139 142 142 141 141 146 145 142 145 6 141 142 140 139 142 141 142 144 140 142 7 145 144 143 142 141 137 145 141 142 141 8 142 145 142 139 138 142 142 141 141 140 9 142 143 141 145 144 139 145 144 146 142 10 142 143 136 139 145 141 143 144 140 142 11 137 141 142 142 139 141 142 139 143 142 12 139 139 138 138 141 143 142 142 144 146 13 139 143 142 142 145 141 141 141 142 142 14 144 144 139 141 142 142 147 142 143 143 15 140 140 140 142 144 140 141 144 142 142 16 141 138 143 141 145 142 137 145 141 140 17 140 141 139 141 142 142 142 139 142 144 18 140 140 139 140 144 142 140 142 144 143 19 140 142 139 142 136 139 144 143 144 141 20 139 140 141 141 138 142 142 146 145 144 21 146 138 143 143 141 143 147 142 145 143 22 138 139 141 141 142 143 146 144 141 141 23 139 142 140 140 140 142 140 144 143 139 24 142 141 143 140 141 140 143 146 142 142Assume that the Normal distribution applies to the data and use a 98% confidence level.a. Find the sample mean and sample standard deviation of the incomes in your data, showing your work. (1) (Your mean should be between 140 and 146 and your sample standard deviation should be around 2.)b. State your null and alternative hypotheses (1) c. Test the hypothesis using a test ratio (1) d. Test the hypothesis using a critical value for a sample mean. (1)e. Test the hypothesis using a confidence interval (1)1252 1takehome081 2/22/08 f. Find an approximate p-value for the null hypothesis. (1)g. On the basis of your tests, is the mean temperature correct in your restaurant?? Why? (1)h. How do your conclusions change if the random sample of 24 temperatures is taken on a day in which only 48 cups cocoa are sold? (2)i. Assume that the Normal distribution does not apply and test to see if the median is 142. Be careful! What should you do with numbers that are exactly 142? (2) [12]j. (Extra Credit) Do a 98% confidence interval for the median. (2)Problem 2: Once again assume that the Normal distribution applies to the data in Problem 1, but that we know that the population standard deviation is 2. Our confidence level remains 98%, but we are now testingthe hypothesis that the mean is below 143 degrees.a. State your null and your alternative hypotheses. (1)b. Find the value of zthat you need for a critical value for a 1-sided test if the confidence level is 98%. You may use a confidence level of 99% if you wish for slightly less credit.c. Find a critical value for the sample mean to test if the mean is below 143 degrees. (1)d. Test the hypothesis that the mean is below 143 degrees using an appropriate confidence interval. (2) e. Using your critical value from 2b, create a power curve for your test. (6) f. Assume that the population standard deviation is 2. How large a sample do you need to get a two-sided 98% confidence interval with an error not exceeding 5.0degrees? (2) [22]Problem 3: According to Doane and Seward about 13% of goods bought at a department store are returned.An organization called Return Exchange will sell you a software product called Verify-1for which it makes the claims below.Verify-1® is quickly operational. And it authorizes returns even quickerVerify-1® identifies fraud and abuse at the point of return before they become liabilities to your brand equity or profits. In stand-alone mode, this easy-to-use, turnkey solution can be operational in 30 days and will reduce your return rate immediately, without disrupting your business or IT configuration. Verify-1® also integrates easily into your existing POS platform.You set the policy, Verify-1® enforces itWith Verify-1®, your returns are dealt with consistently utilizing advanced statistical modeling in combination with state return laws and your existing return policies. At the point of return, using the customer’s driver’s license or other valid identification, Verify-1® automatically checks prior return behavior and authorizes or declines the transaction. Customers identified as risks for presenting fraudulent returns are declined, while legitimate returns are speedily accepted.You take a sample of nitems and
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