Let Z1,...,Zn be independently normally distributed with common variance 2 and means E(Zi) = i(i = 1,...,s),

Let Z1,...,Zn be independently normally distributed with common variance σ2 and means E(Zi) = ζi(i = 1,...,s), E(Zi)=0(i = s+1,...,n).

There exist UMP unbiased tests for testing ζ1 ≤ ζ0 1 and ζ1 = ζ0 1 given by the rejection regions Z1 − ζ0 6 1 n i=s+1 Z2 i /(n − s)
> C0 and |Z1 − ζ0 1 |
6 n i=s+1 Z2 i /(n − s)
> C.
When ζ1 = ζ0 1 , the test statistic has the t-distribution with n − s degrees of freedom.