1. Here is the power function for a particular test about p = fraction of beverage orders for Coke in a restaurant. Suppose that p 0 = 0.3.

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1 Recitation Assignment due Exam 5 coverage: Most of the exam questions go to material from chaters 16 and 20, but some may touch on suorting material from chater 15. Exect to see questions relating to and couched in language develoed in lectures and recitation assignments. 1. Here is the ower function for a articular test about = fraction of beverage orders for Coke in a restaurant. Suose that 0 = 0.3. a. Determine the null hyothesis H 0 and the alternative hyothesis H A. b. Determine the robability of tye one error a (i.e. chance the test will reject the null hyotheses if = 0 ). This a is secified by the exerimenter constructing the test and is termed the "significance level of the test." One also seaks of the "a level of the test."

2 2 rec nb "significance level of the test." One also seaks of the " level of the test." c. Determine the robability of tye two error (chance the test will fail to reject H 0 ) if = d. If Coke sales are running at 10% is there much chance of rejecting the null hyothesis using the test with this ower curve? e. If Coke sales are running at 40% is there much chance of rejecting the null hyothesis using the test with this ower curve? 2. Two ower curves are overlaid below. Both have the same significance level a and the same null and alternative hyotheses. a. Identify the null and alternative hyotheses, a. Which test is better? b. Is this a one sided or two sided test?. c. By hand, draw in the ideal ower curve based on erfect information.

3 rec nb 3 c. By hand, draw in the ideal ower curve based on erfect information. 3. Two ower curves are overlaid below. Both have the same significance level a and the same null and alternative hyotheses. a. Identify the null and alternative hyotheses. Is the test one or two sided? b. Determine the significance level a. c. By hand, draw in the ideal ower curve based on erfect information. d. Which is the ower curve of the better test? e. If = 0.55 what is the robability of rejecting the null hyothesis uner each of the two tests?

4 4 rec nb 4. An election its Reublican candidate against a Democrat candidate. Denote by the fraction of the electorate currently favoring the Reublican. Here is the ower curve for a ossible test (based on a random samle of 800 voters) of the null hyothesis that the Reublican candidate is leading in votes. The alternative hyothesis is the Reublican candidate is not leading in votes. a. Identify the null and alternative hyotheses on the axis and label the axis also as "fraction favoring Reublican." b. What is the value of the robability a of tye one error (concluding that the Democrat is ahead when actually the Reublican is at 50%)? c. Which arty should comlain they would be treated unfairly by the test?

5 rec nb 5 d. If the Reublican share of the vote is = 5 what is the robability that the test will reject the null hyothesis? e. If you could devise a test with the same shae of ower curve what would you do to the curve to make the associated test fair to both arties? 5. A statistical test of the null hyothesis that wine orders are running at 30% versus the alternative that they run at greater than 30% will be based on a random samle of 400 orders and a =. In the samle there are 150 wine orders. a. With the given information what is the value of `? b. With the given information what is the value of 0? c. Determine the value of SD( 0 ) = 0 H1-0 L n. SD(`) is the notation used for this in your textbook but let's not confuse it with the estimated SD of ` (as used in CI) which is ` H1-`L n. d. Determine the value of the test statistic denoted z = `- 0 SDH 0 L. e. Determine the -value = P(Z > z) where z is the value of the test statistic (d). f. The test rejects H 0 if -value < a. What action is taken by the test? Is it to reject the null hyothesis or to fail to reject the null hyothesis?

6 6 rec nb g. Had there been 100 wine orders in the samle would there have been any reason to carry out the hyothesis test? 6. A statistical test of the null hyothesis that wine orders are running at 30% versus the alternative that they run at some value OTHER THAN 30% will be based on a random samle of 400 orders and a =. In the samle there are 90 wine orders. a. With the given information what is the value of `? b. With the given information what is the value of 0? c. Determine the value of SD( 0 ) = 0 H1-0 L n. d. Determine the value of the test statistic denoted z = `- 0 SDH 0 L. e. Determine the -value = 2 P( Z > z ) where z is the value of the test statistic (d). The reason for the 2 is that this two sided test rejects when ` is either too large or too small so both tail robabilities have to be included. f. What action is taken by the test? Is it to reject the null hyothesis or to fail to reject the null hyothesis? g. Had there been 100 wine orders in the samle would there have been any reason to carry out the hyothesis test?