A company has a manufacturing plant for making widgets in Lexington, Kentucky but needs to expand its production capabilities. The company builds a new manufacturing plant in Brookings, South Dakota. A process engineer wants to see if widgets made on the new manufacturing line in South Dakota can be shown to have an increased performance lifetime over those made on the legacy line in Kentucky. To test this claim, she takes 50 of the widgets made in South Dakota and finds they have an average lifetime of 1,586 hours with a standard deviation of 35 hours. She also takes 50 of the widgets made in Kentucky and finds they have an average lifetime of 1,577 hours and a standard deviation on 39 hours. Is the performance of the widgets from the new line superior at the 0.05 level of significance? A new developmental polymer for making textile fibers is purported to have an increase of 1.5 N/tex in tenacity over the competitor. In a sample of 8 of the developmental fibers, the mean tenacity is 9.67 N/tex with a standard deviation of 1.81 N/tex. In a sample of 10 of the competitor fibers, the mean tenacity is 7.43 N/tex with a standard deviation of 1.48 N/tex. Is the claim of the improved tenacity of the developmental fiber true at the 0.05 level of significance? Construct a 95% confidence interval for the previous problem regarding the difference in tenacity of the experimental polymer fibers over the competitor fibers