# ttest where the difference in the null hypothesis is not 0: non-centrality parameter?

I have problems to understand when a non-centrality parameter is relevant. As far as I understood, if the t distribution is not centered on 0, a non-central t distribution is used. For example, in power analysis, it is assumed that an effect is present, therefore, a non-central t is used to calculate the probabilities. Now to my application: If I carry out a t test where the null hypothesis is not that the difference is 0, eg:

H0: μ1 − μ2 ≤ 3

H1: μ1 − μ2 > 3

Do I need a noncentral t distribution here to calculate the pvalue for H0? Is this non-centrality parameter = 3?

Does anyone have an good source for the relevance of non-central distributions in social sciences?

• Under the null hypothesis $\mu_1 - \mu_2 = 3$, what's the expectation of $\bar{X}_1 - \bar{X}_2 - 3$ (where $\bar{X}_j$ is the mean of the $j$th sample)? – Scortchi Jan 16 '17 at 11:35
• I suppose 3? Do you need more background info? – 00schneider Jan 16 '17 at 11:37
• Just a hint - I think your question's clear enough. (The answer's not 3 though.) – Scortchi Jan 16 '17 at 11:54
• Then I suppose it is 0 and I do not need a noncentral t? Is my understanding of noncentral t, as posted in the original post, correct? – 00schneider Jan 16 '17 at 14:17

E(X1 - X2 -3) = E(X1) - E(X2) - 3 = u1 - u2 -3 = 0