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What is z-test and t-test ?

A Z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution. Z-test tests the mean of a distribution in which we already know the POPULATION variance σ2 . Because of the central LIMIT theorem, many test statistics are approximately normally DISTRIBUTED for large samples. For each significance level in the confidence interval, the Z-test has a single critical value (for example, 1.96 for 5% two tailed) which makes it more convenient than the Student's t-test which has separate and different critical values for each sample size (for different sample size, it would have different degree of freedom, which may determine the value of the critical values). Therefore, many statistical tests can be conveniently performed as approximate Z-tests if the sample size is large or the population variance is KNOWN. If the population variance is unknown (and therefore has to be estimated from the sample itself) and the sample size is not large (n < 30), the Student's t-test may be more appropriate.

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What conditions are necessary in order to use the z-test to test the difference between two population means?

The samples must be randomly SELECTED, each population has a normal distribution with a known standard deviation, the samples must be independent. All population means are equal.

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