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Let X₁, ..., X_n be i.i.d. from N(µ, σ²) with unknown µ and σ². Consider the problem of testing H₀; µ = 0 versus H₁; µ ≠ 0 and the group of transformations gc(X_i) = cX_i, c ≠ 0.

(a) Show that the testing problem is invariant under G.

(b) Show that the one-sample two-sided t-test in §6.2.3 is a UMPI test.

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