Problems 1 Let Y1, Y2, Y3, and Y4 be independent, identically distributed random variables from a population with mean and variance o. Let Y variables. = (Y+ Y+Y3+ Y4) denote the average of these four random 4 (i) What are the expected value and variance of Y in terms of and ? (ii) Now, consider a different estimator of : 1 W==Y + 8 1 8 1 Y+Y+Y 4 1 2 This is an example of a weighted average of the Y. Show that W is also an unbiased estimator of . Find the variance of W. (iii) Based on your answers to parts (i) and (ii), which estimator of m do you prefer, Y or W? non si tavos 7 The new management at a bakery claims that workers are now more productive than they were under U bat old management, which is why wages have "generally increased." Let W; be Worker i's wage under the old management and let W be Worker i's wage after the change. The difference is D = W - Wb. Assume that the D; are a random sample from a Normal (u, 2) distribution. . x(i) Using the following data on 15 workers, construct an exact 95% confidence interval for (ii) Formally state the null hypothesis that there has been no change in average wages. In particular, what is E(D;) under Ho? If you are hired to examine the validity of the new management's claim, what is the relevant alternative hypothesis in terms of = E(D;)? (iii) Test the null hypothesis from part (ii) against the stated alternative at the 5% and 1% levels. (iv) Obtain the p-value for the test in part (iii). Worker Wage Before Wage After 1 8.30 9.25 2 9.40 9.00 3 9.00 9.25 4 10.50 10.00 5 11.40 12.00 678 8.75 9.50 10.00 10.25 9.50 9.50 9 10.80 11.50 10 12.55 13.10 11 12.00 11.50 12 8.65 9.00 13 7.75 7.75 14 11.25 11.50 15 12.65 13.00