We are to test
H0: θ = θ0 versus H1: θ > θ0,
where θ is the mean of one of two normal distributions and θ0 is a fixed but arbitrary value of θ. We observe the random variable X with distribution

a. Show that the test given by
reject H0 if X > θ0 + zασ,
where σ = 1 or 10 depending on which population is sampled, is a level a test. Derive a 1 - α confidence set by inverting the acceptance region of this test.
b. Show that a more powerful level a test (for α > p) is given by
reject H0 if X > θ0 + z(α-p))/(1-p) and σ = 1; otherwise always reject H0.
Derive a 1 - α confidence set by inverting the acceptance region of this test, and show that it is the empty set with positive probability. (Cox's Paradox states that classic optimal procedures sometimes ignore the information about conditional distributions and provide us with a procedure that, while optimal, is somehow unreasonable; see Cox 1958 or Cornfield 1969.)

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