What are the critical values for a two-tailed test with a 0.01 level of significance when n is large and the population standard deviation is known?

For a two-tailed test we need to halve 0.01, giving 0.005. Then we need to sutract 0.005 from 0.5, giving 0.5 - 0.005 = 0.495. Looking up the p value of 0.495 in an inverse normal probability table we obtain the value z = 2.5758. Therefore the critical values are -2.5758 and 2.5758.

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