Let ${X_n : n=1,2,cdots}$ be iid random variables with distribution $P_{ heta}, heta in H$ where $H$ is some abstract parameter space. We consider a sequential confidence interval I for the mean $mu = mu( heta)$ of $P_{ heta}$ satisfying $P_{ heta}(mu in I) geq 1-alpha$ and $P_{ heta}(mu-delta(mu) in I) leq eta$ for all $ heta in H$ for any given an imprecision real valued function $delta(mu) > 0$ and error probabilities $0 < alpha, eta < 1$. A one-sided sequential confidence interval is constructed under some restriction of the family {P_{ heta} : heta in H}$ and the imprecision function $delta$. This is extended to the two-sided cases.