Let k denote the number of successes observed in a sequence of n independent Bernoulli trials, where p = P(success).

(a) Show that the critical region of the likelihood ratio test of H₀: p = 1/2 versus H₁: p ≠ 1/2 can be written in the form k・ ln(k) + (n − k) ・ ln(n − k) ≥ λ^∗∗

(b) Use the symmetry of the graph of f(k) = k ・ ln(k) + (n − k)・ ln(n − k) to show that the critical region can be written in the formwhere c is a constant determined by α.

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2 c