# What are the two 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?

When the sample size `n` is ` ` "large" (some people use  `n>20` as a rough rule of thumb, but this is a subjective choice), whatever the distribution of the measure in the population is we can appeal to the Central Limit Theorem and assume that the mean of our sampled data is Normal/Gaussian with standard deviation `sigma/sqrt(n)` where ` ` `sigma` is the population standard deviation. Note that if the distribution of the measure in the population is Normal and the population standard deviation is known, the sampling mean will follow a Normal distribution even for small `n`.

To conduct a two-tailed test at the 1% level of significance, we need to consider the 0.5 and 99.5 percentiles of the sampling distribution (1/2% in each tail)

Now, the 0.5 and 99.5 percentiles of the Normal distribution are `pm 2.576`

Therefore, the critical values of a two-tailed test for a "large" sample of data are

`pm (sqrt(n)/sigma) 2.576`

If the test statistic calculated from the sample falls outside of the range `[-2.576,2.576](sqrt(n)/sigma)`, then the null hypothesis is rejected at the 1% level of significance.

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