Assumptions: Observations are independent of one another. The data come from a normal distribution x1i ∽ N(μ1,σ12) and x 2i ∽ N(μ2,σ22), with σ 12 and σ 22 known or large n1 and large n2 with σ12 and σ 22 known. Note: This test will not be stressed as again the population variance is almost always unknown in real life. In addition, software such as SPSS, does not even offer this option for the latter reason.
Typically for two samples z-tests the null hypothesis has μ1 = μ2. The null hypothesis can be written as μ1 = μ2 + δ, where δ is some constant representing the quantity difference between the population means. In this section and other sections difference between μ1 and μ2, is assumed equal to zero, δ = 0.
A (1 - α)100% confidence interval for μ is,
