(i) Let X1,...,Xn be a sample from N(, 2), and consider the problem of deciding between 0...

(i) Let X1,...,Xn be a sample from N(ξ, σ2), and consider the problem of deciding between ω0 : ξ < 0 and ω1 : ξ ≥ 0. If ¯x = xi/n and C = (a1/a0)

2/n, the likelihood-ratio procedure takes decision d0 or

d, as

√nx¯ (xi − x¯)2 < k or > k, where k = √C − 1 if C > 1 and k = (1 − C)/C if C < 1.

(ii) For the problem of deciding between ω0 : σ<σ0 and ω1 : σ ≥ σ0 the likelihood ratio procedure takes decision d0 or

d, as (xi − x¯)
2 nσ2 0 < or > k, where k is the smaller root of the equation Cx = ex−1 if C > 1, and the larger root of x = Cex−1 if C < 1, where C is defined as in (i).
Section 1.8