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(θ,1). (a) Define a statistic T = g(X,θ) whose distribution does not
depend on θ and use it as a pivotal statistic to obtain a 100(1−α)% confidence interval for
θ. (b) If X = 1087 pounds was the measured resistance, obtain a 95% confidence interval
for θ.