On the surface of the planet Mars, the vehicle Curiosity made carefully selected probes to measure the hardness of Martian soil. These probes were considered to be risky because of the non-negligible risk that the instrument to make the measurement would break. In a particular region under study, the resistance of the surface to the probe has been modeled as a Weibull variable. Because of risk considerations, only a single measurement X was taken, with X W (, 1).

(a) Let U = F(X). Prove that U U(0,1).

(b) Using (a) define a pivotal quantity Q(X,) and obtain an equal-tailed interval of (100-a)% for .

(c) Obtain a minimal length (100-a)% confidence interval for .

(d) If X = 1087 pounds was the measured resistance, obtain a 95% confidence interval for .