Two different formulations of an oxygenated motor fuel are being tested to study their road octane numbers. The variance of road octane number for formulation 1 is σ²₁ = 1.5, and for formulation, 2 it is σ²₂ = 1.2. Two random samples of size n₁ = 15 and n₂ = 20 are tested, and the mean road octane numbers observed are x̅₁ = 89.6 and x̅₂ = 92.5. Assume normality.

(a) If formulation 2 produces a higher road octane number than formulation 1, the manufacturer would like to detect it. Formulate and test an appropriate hypothesis using α = 0.05. What is the P-value?

(b) Explain how the question in part (a) could be answered with a 95% confidence interval on the difference in mean road octane number.

(c) What sample size would be required in each population if you wanted to be 95% confident that the error in estimating the difference in mean road octane number is less than 1?