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The main objective of most clinical trials is to estimate the effect of some treatment compared to a control condition. We define the signal‐to‐noise ratio (SNR) as the ratio of the true treatment effect to the SE of its estimate. In a previous publication in this journal, we estimated the distribution of the SNR among the clinical trials in the Cochrane Database of Systematic Reviews (CDSR). We found that the SNR is often low, which implies that the power against the true effect is also low in many trials. Here we use the fact that the CDSR is a collection of meta‐analyses to quantitatively assess the consequences. Among trials that have reached statistical significance we find considerable overoptimism of the usual unbiased estimator and under‐coverage of the associated confidence interval. Previously, we have proposed a novel shrinkage estimator to address this “winner's curse.” We compare the performance of our shrinkage estimator to the usual unbiased estimator in terms of the root mean squared error, the coverage and the bias of the magnitude. We find superior performance of the shrinkage estimator both conditionally and unconditionally on statistical significance.