Supercavitation is a propulsion technology for undersea vehicles that can greatly increase their speed. It occurs above approximately 50 meters per second when pressure drops sufficiently to allow the water to dissociate into water vapor, forming a gas bubble behind the vehicle. When the gas bubble completely encloses the vehicle, supercavitation is said to occur. Eight tests were conducted on a scale model of an undersea vehicle in a towing basin with the average observed speed x̅ = 102.2 meters per second. Assume that speed is normally distributed with known standard deviation σ = 4 meters per second.

(a) Test the hypothesis H₀: μ = 100 versus H₁: μ <100 using α = 0.05.

(b) What is the P-value for the test in part (a)?

(c) Compute the power of the test if the true mean speed is as low as 95 meters per second.

(d) What sample size would be required to detect a true mean speed as low as 95 meters per second if you wanted the power of the test to be at least 0.85?

(e) Explain how the question in part (a) could be answered by constructing a one-sided confidence bound on the mean speed.