Consider the hypothesis test H 0 : 1 = 2 against H 1 : 1 2 with known variances

Consider the hypothesis test H₀: μ₁ = μ₂ against H₁: μ₁ ≠ μ₂ with known variances σ₁ = 10 and σ₂ = 5. Suppose that sample sizes n₁ = 10 and n₂ = 15 and that x̅₁ = 4.7 and x̅₂ = 7.8. Use α = 0.05. 

(a) Test the hypothesis and find the P-value.

(b) Explain how the test could be conducted with a confidence interval.

(c) What is the power of the test in part (a) for a true difference in means of 3?

(d) Assume that sample sizes are equal. What sample size should be used to obtain β = 0.05 if the true difference in means is 3? Assume that α = 0.05.