Ii. [15 pts} Sample size, power, and two sample tests. Exxon Mobil two years ago pledged 1 million dollars to lobby for a carbon tax. Suppose you want to see how a carbon tax affects the environmental performance of companies. You will randomize companies into two groups one group will have a carbon tax, the other will not have a carbon tax. lnrl'ter a year, you measure an environmental performance metric [EPM]: for each company which takes values from D to 100, 1130 being higher environmental performance. ll'ou want to see ifthe mean EPlv'l of companies with a carbon tax is higher after a year than the mean EPlv'I for nontaxed companies. We investigated the relationship bethEn power, alpha, sample sizes, and standard deviations in class. In this problem. you'll use an online sample sizefpower calculator: Here are your BSSLIH'IIJtiDI'IS - I Mean EPiv'l score of taxed companies is estimated to be TU {mu 1} a Mean EPM score of non-taxed companies is estimated to be 50 [mull - The standard deviation of EPlv'l scores for both groups is assumed to be 25lsigma} I We want alpha = .EIS and the power to be .30. Assume a two-sided test {i.e. we're using a twosided alternative hypothesis which means the carbon tax could make EPM go up or down] a. Use the assumptions above and the indicated website to calculate the estimated sample size needed for each group in order to detect a statistically signicant difference. Report this value. b. Leaving all other values the same, reduce alpha to [1.01. What is the new sample size in each group? Explain in a sentence why a larger sample size is expected c. Return alpha to 0.1235. Assume that the carbon tax doesn't work as well as you think - in fact, the mean EPlvl is only 60 in the carbon tax group. Now what is the estimated sample size required in each group? Explain in a sentence why you expect an increase in sample size. d. Return all values to the original values. Now, assume you decide to pick similar companies in order to reduce the expected standard deviation to 20. What is the new sample size required? Explain in a sentence why the smaller sample size is expected. e. Often, budgets and time determine sample size. Using all of the original assumptions {except for power}; assume you can afford 113 companies in each group. Click on the button that says ("Calculate Power (for specified Sample Size)\". What is the power of your test for a sample size of 10 in each group? Explain in a sentence why you expect lower power. Inference for Means: Comparing Two Independ (To use this page, your browser must recognize JavaScript.) Choose which calculation you desire, enter the relevant population values for for each sample). You may also modify a (type I error rate) and the power, if O Calculate Sample Size (for specified Power) O Calculate Power (for specified Sample Size) Enter a value for mul: 70 Enter a value for mu2: 50 Enter a value for sigma: 25 O 1 Sided Test O2 Sided Test Enter a value for a (default is .05): 05 Enter a value for desired power (default is .80): 0.43 The sample size (for each sample separately) is: 10Inference for Means: Comparing Two Independent Samples (To use this page, your browser must recognize JavaScript.) Choose which calculation you desire, enter the relevant population values for mul (mean of popul for each sample). You may also modify a (type I error rate) and the power, if relevant. After makin O Calculate Sample Size (for specified Power) O Calculate Power (for specified Sample Size) Enter a value for mul: 70 Enter a value for mu2: 50 Enter a value for sigma: 25 O 1 Sided Test 2 Sided Test Enter a value for a (default is .05): 05 Enter a value for desired power (default is .80): .80 The sample size (for each sample separately) is: 25