Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be σ₁ = 0.10 fluid ounces and σ₂ = 0.15 fluid ounces for the two machines, respectively. Two random samples of n₁ = 12 bottles from machine 1 and n₂ = 10 bottles from machine 2 are selected, and the sample mean fill volumes are x̅₁ = 30.87 fluid ounces and x̅₂ = 30.68 fluid ounces. Assume normality.

(a) Construct a 90% two-sided confidence interval on the mean difference in fill volume. Interpret this interval.

(b) Construct a 95% two-sided confidence interval on the mean difference in fill volume. Compare and comment on the width of this interval to the width of the interval in part (a).

(c) Construct a 95% upper-confidence interval on the mean difference in fill volume. Interpret this interval.

(d) Test the hypothesis that both machines fill to the same mean volume. Use α = 0.05. What is the P-value?

(e) If the β-error of the test when the true difference in fill volume is 0.2 fluid ounces should not exceed 0.1, what sample sizes must be used? Use α = 0.05.