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A z-test is any statistical test otherwise known as a hypothesis for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution (“Z-test,” 2018). A one-sample location test, two-sample location test, paired difference test and maximum likelihood estimate are examples of tests that can be conducted as z-tests. Z-tests are closely related to t-tests, but a z-test assumes the standard deviation is known and the sample size is large.

The reason a z-test would be preferred over a t-test is when an investor wishes to test whether the average daily return of a stock is greater than 1%. A simple random sample of 50 returns is calculated and has an average of 2%. Assume the standard deviation of the returns is 2.50%. Therefore, the null hypothesis is when the average, or mean, is equal to 3%. The alternative hypothesis is whether the mean return is greater than 3%. Assume an alpha of 0.05% is selected with a two-tailed test. Consequently, there is 0.025% of the samples in each tail, and the alpha has a critical value of 1.96 or -1.96. If the value of z is greater than 1.96 or less than -1.96, the null hypothesis is rejected. Nuisance parameters should be known or estimated with high accuracy. Z-tests focus on a single parameter, and treat all other unknown parameters as being fixed at their true values (“Z-test,” 2018). Z-tests are not commonly used because it is not as straightforward or easy to use compared to t-tests.

References

Z-test. (2018). In Wikipedia. Retrieved October 8, 2018, from https://en.wikipedia.org/wiki/Z-test

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