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The increasing interest in subpopulation analysis has led to the development of various new trial designs and analysis methods in the fields of personalized medicine and targeted therapies. In this paper, subpopulations are defined in terms of an accumulation of disjoint population subsets and will therefore be called composite populations. The proposed trial design is applicable to any set of composite populations, considering normally distributed endpoints and random baseline covariates. Treatment effects for composite populations are tested by combining $p$-values, calculated on the subset levels, using the inverse normal combination function to generate test statistics for those composite populations. The family-wise type I error rate for simultaneous testing is controlled in the strong sense by the application of the closed testing procedure. Critical values for intersection hypothesis tests are derived using multivariate normal distributions, reflecting the joint distribution of composite population test statistics under the null hypothesis. For sample size calculation and sample size re-calculation multivariate normal distributions are derived which describe the joint distribution of composite population test statistics under an assumed alternative hypothesis. Simulations demonstrate the strong control of the family-wise type I error rate in fixed designs and re-calculation designs with blinded sample size re-calculation. The target power after sample size re-calculation is typically met or close to being met.