The new management at a bakery claims that workers are now more productive than they were under old management, which is why wages have “generally increased.” Let W^b_i be Worker i’s wage under the old management and let W^a_i be Worker i’s wage after the change. The difference is D_i ≡ W^a_i –W^b_i.

Assume that the D_i are a random sample from a Normal (µ, s²) distribution.

(i) Using the following data on 15 workers, construct an exact 95% confidence interval for µ.

(ii) Formally state the null hypothesis that there has been no change in average wages. In particular, what is E(D_i) under H₀? If you are hired to examine the validity of the new management’s claim, what is the relevant alternative hypothesis in terms of µ = E(Di)?

(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).

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Worker Wage Before Wage After 1 8.30 9.25 2 9.40 9.00 3 9.00 9.25 10.50 10.00 5 11.40 12.00 8.75 9.50 7 10.00 10.25 8 9.50 9.50 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 4.