A manufacturing company considers to use one of two types of materials based on the breaking strength. The sigma of the two materials are equal, and the value is 1.0 pounds per square inch (psi). Two random samples of different sizes are taken: n1= 10, and n2= 12. The sample means are calculated to be x-bar1= 162.5 and x-bar2= 155.0. The company will not use material type 1 unless its mean breaking strength exceeds that of material type 2 by at least 10 psi.

a) Should the company use material type 1 based on the sample information (the decision will be made at a significance level of 0.05). (Use Hypothesis Testing; Specify all steps, and the conclusion regarding Ho and a verbal statement).

b) Calculate the P-value for part a).

c) What is the 95% CI on the difference in means, assuming that the difference in means is indeed 12 psi?

d) If alpha = 0.05, what is the power of the test?

e) The company wishes to find out if the sample sizes used (n1=10 and n;=12) are adequate to detect a difference of 12 psi. They will decide this based on calculating the needed sample size (considering equal sample sizes) using the data of the problem and the values calculated in the parts a) to c). If this sample size is smaller than the sizes used then the sample sizes 10 and 12, are adequate. Calculate the needed sample size (n), and conclude if the initial sample sizes are adequate. (Make a brief verbal statement).