• Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be σ₁ = 0.10 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 ₁ = 30.61 and 2 = 30.34 fluid ounces. Assume normality. a) Construct a 90% two-sided Cl on the mean difference in fill volume. Interpret this interval. b) Construct a 95% two-sided Cl 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 bound on the mean difference in fill volume. Interpret this interval. Reconsider previous problem a) Test the hypothesis that both machines fill to the same mean volume. Use the P-value approach. (b) If a 0.05 and the ẞ-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 0.05.