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The quantum entangled J=ψ → ΣþΣ¯ − pairs from ð1.0087 0.0044Þ × 1010 J=ψ events taken by the BESIII detector are used to study the nonleptonic two-body weak decays Σþ → nπþ and Σ¯ − → n¯π−. The CP-odd weak decay parameters of the decays Σþ → nπþ (αþ) and Σ¯ − → n¯π− (α¯−) are determined to be 0.0481 0.0031stat 0.0019syst and −0.0565 0.0047stat 0.0022syst, respectively. The decay parameter α¯− is measured for the first time, and the accuracy of αþ is improved by a factor of 4 compared to the previous results. The simultaneously determined decay parameters allow the first precision CP symmetry test for any hyperon decay with a neutron in the final state with the measurement of ACP ¼ ðαþ þ α¯−Þ=ðαþ − α¯−Þ ¼ −0.080 0.052stat 0.028syst. Assuming CP conservation, the average decay parameter is determined as hαþi¼ðαþ − α¯−Þ=2 ¼ −0.0506 0.0026stat 0.0019syst, while the ratios αþ=α0 and α¯−=α¯ 0 are −0.0490 0.0032stat 0.0021syst and −0.0571 0.0053stat 0.0032syst, where α0 and α¯ 0 are the decay parameters of the decays Σþ → pπ0 and Σ¯ − → p¯ π0, respectively.