This test is used to determine whether the mean values obtained from two independent samples which provide the best estimate of two population means, μ1 and μ2, differ by a specified amount d, when the population standard deviation, σ, is known (or where a good estimate, s, is available based on a sample size n>30).

Assumptions: The sample is random and the population is Normally distributed

Hypothesis: H0: μ1-μ2=d; H1: μ1-μ2≠d

Test: Compute the z-statistic:

z is distributed approximately N(0,1). If d=0 then the test is one of equality of the two mean values. If the two populations are finite, of size N1 and N2, the standard error components of the z-statistic should be replaced with the adjusted expressions:

Note: if the samples are actually paired rather than independent, the difference between the pairs of values can be computed and then the mean of these differences tested using the z-test for a single mean.