Z-test in simple words

The z-test is a statistical test used to determine whether two population means are significantly different when the population's standard deviation is known. This test is widely used in many fields, including business, science, and engineering. The z-test is a parametric test, which means that it assumes the data follows a normal distribution. In order to perform the z-test, the researcher needs to know the mean and standard deviation of the population. The z-test compares the difference between the sample means to the difference that would be expected if the two-population means were equal. The formula for the z-test is:

z = (x1 - x2) / (σ / sqrt(n))

where x1 and x2 are the sample means, σ is the standard deviation of the population, and n is the sample size.

To perform the z-test, the researcher first sets a level of significance, which is usually denoted by α. The level of significance is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. The most commonly used level of significance is 0.05, which means there is a 5% chance of making a Type I error.

The null hypothesis for the z-test is that there is no significant difference between the two population means. The alternative hypothesis is that there is a significant difference between the two population means. The researcher calculates the test statistic, which is the z-score and compares it to the critical value obtained from a standard normal distribution table. If the test statistic is greater than the critical value, the null hypothesis is rejected and the alternative hypothesis is accepted.

The z-test has several advantages over other statistical tests. First, it is easy to understand and interpret the results. Second, it is a powerful test that can detect small differences between population means. Third, it can be used to test hypotheses about population means when the sample size is large.

However, the z-test also has some limitations. First, it requires that the standard deviation of the population is known, which is often not the case. Second, it assumes that the data follows a normal distribution, which may not always be the case. Third, it is a one-tailed test, which means that it can only detect differences in one direction.

In conclusion, the z-test is a powerful statistical test that can be used to determine whether two population means are significantly different when the population's standard deviation is known. However, it also has some limitations and should be used with caution. Researchers should carefully consider the assumptions and limitations of the z-test before using it in their research.