Two different types of injection-molding machines are used to form plastic parts. A part is considered defective if it has excessive shrinkage or is discolored. Two random samples, each of size 300, are selected, and 15 defective parts are found in the sample from machine 1, and 8 defective parts are found in the sample from machine 2.

(a) Is it reasonable to conclude that both machines produce the same fraction of defective parts, using α = 0.05? Find the P-value for this test.

(b) Construct a 95% confidence interval on the difference in the two fractions defective.

(c) Suppose that p₁ = 0.05 and p₂ = 0.01. With the sample sizes given here, what is the power of the test for this two-sided alternate?

(d) Suppose that p₁ = 0.05 and p₂ = 0.01. Determine the sample size needed to detect this difference with a probability of at least 0.9.

(e) Suppose that p₁ = 0.05 and p₂ = 0.02. With the sample sizes given here, what is the power of the test for this two-sided alternate?

(f) Suppose that p₁ = 0.05 and p₂ = 0.02. Determine the sample size needed to detect this difference with a probability of at least 0.9.