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

(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 C x = ex−1 if C > 1, and the larger root of x = Cex−1 if C < 1, where C is defined as in (i)