A sequential procedure is given for deciding to which of $k$ non-overlapping intervals the unknown mean $\theta$ belongs which satisfies the requirement that the probability of making an incorrect decision is less than some preassigned value $\alpha$. The sequential procedure is worked out explicitly for the following two cases: (1) when $\theta$ is the mean of a normal distribution with a known variance, and (2) when $\theta$ is the mean of a normal distribution with an unknown variance. A brief discussion is also given of a related but apparently new problem, to find a sequential procedure which will simultaneously select one of the $k$ intervals and also yield a confidence interval for $\theta$ of a specified width.
"A Sequential Decision Procedure for Choosing One of $k$ Hypotheses Concerning the Unknown Mean of a Normal Distribution." Ann. Math. Statist. 34 (2) 549 - 554, June, 1963. https://doi.org/10.1214/aoms/1177704167