Someone wrote to me asking to see the scratch work for the SEV calculations. (See June 14 post, also LSE problem set.) I’ll just do the second one:
What is the Severity with which (μ<3.29) passes the test T+ in the case where σx = 2? We have that the observed sample mean M is 1.4, so
SEV (μ < 3.29) = P( test T+ yields a result that fits the 0 null less well than the one you got (in the direction of the alternative); computed assuming μ as large as 3.29)
SEV(μ < 3.29) = P(M >1.4; μ >3.29) > P(Z > (1.4 -3.29)/2)) * = P(Z > -1.89/2) = P(Z > -.945 ) ~ .83
*We calculate this at the point μ = 3.29, since the SEV would be larger for greater values of μ.
That’s quite a difference from the power calculation of .5, calculated in the usual way of a discrepancy detect size (DDS) analysis.
NEW PROBLEM: You want to make an inference that passes with high SEV, say you want SEV(μ < μ’) = .99, with the same (statistically insignificant) outcome you got from the second case of test T+ as before (σx = 2). What value for μ’ can you infer μ < μ’ with a SEV of .99?