A certain wastewater treatment method is supposed to neutralize the wastewater so that the mean pH is 7. Measurements of pH will be made on seven specimens of treated wastewater, and a test of the hypotheses H0 :μ = 7 versus H1 :μ ( 7 will be made using the Student's t test (Section 6.4). Assume that the true mean is μ = 6.5, the pH measurements are normally distributed with meanμand standard deviation 0.5, and the test is made at the 5% level.
a. Let X1, . . . , X7 denote the pH measurements, let X̅ denote their mean, and let s denote their sample standard deviation. What is the test statistic? For what values of the test statistic will H0 be rejected?
b. Generate 10,000 samples X∗1, . . . , X∗7 from the true distribution of pH measurements. For each sample, compute the test statistic and determine whether H0 is rejected. Estimate the power of the test?