The Sign Test That Cannot Reach Five Percent
Seventeen subjects are measured before and after a treatment. Four of them show no change and are dropped, leaving 13 non-zero differences. Under $H_0$ (median difference zero) the number of positive differences is Binomial(13, 1/2). The two-sided sign test rejects when the count of positive differences lies in a symmetric region $\{0,\dots,c\}\cup\{13-c,\dots,13\}$, where $c$ is chosen as large as possible subject to the true size being at most 0.05. What is the exact true size (type I error probability) of this test? Give a reduced fraction.
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Source: Original problem written for AxiomIQ.