Nice post. I don’t know about you, but it seems that all the definitions and information regarding when to use the z-test and t-test are the same. There seems to be no discrepancy regarding this. The z-test is for sample size greater then 30 and when the standard deviation is known. Otherwise the t-test is used. I cannot find if there are any exceptions to this rule. Have you come across any exceptions to the t-test and z-test rules of when to use each hypothesis ?
While the general principles you mentioned about the use of the z-test and t-test are indeed commonly followed, there are some exceptions to these rules. The z-test and t-test are statistical hypothesis tests used to make inferences about population parameters based on sample data. The choice between the two tests depends on various factors, including the sample size, whether the population standard deviation is known, and the desired level of confidence.
Firstly, let’s consider the scenario where the sample size is less than 30. It is often suggested that the t-test should be used in this situation. However, this guideline assumes that the population follows a normal distribution. If the population distribution is not normal, the t-test may not provide accurate results, and alternative non-parametric tests, such as the Wilcoxon rank-sum test or the Mann-Whitney U test, may be more appropriate.
Secondly, while the z-test is typically used when the population standard deviation is known, there are instances where the t-test can still be used even if the population standard deviation is known. This can occur when the sample size is large, typically greater than 30. In such cases, the t-distribution closely approximates the standard normal distribution, and the results of the t-test and z-test will be quite similar. Hence, if the sample size is large and the population standard deviation is known, both the t-test and z-test can be used interchangeably.
On the other hand, there are scenarios where the z-test can be employed even if the population standard deviation is unknown. This happens when the sample size is sufficiently large, generally greater than 30. In such cases, the sample standard deviation can be used as an estimate of the population standard deviation, and the central limit theorem ensures that the distribution of the sample mean is approximately normal. Consequently, the z-test can be applied instead of the t-test.
Moreover, exceptions to these rules may also arise when dealing with specific types of data or research designs. For instance, when analyzing paired data, such as in a paired t-test, the rules for using the t-test differ from those for independent samples. Similarly, in the case of one-sample tests, where the sample mean is compared to a known population mean, the t-test may be used even when the sample size is small, as long as the data meet the assumptions for using the t-distribution.
Furthermore, it is worth noting that the z-test and t-test are both parametric tests, meaning they assume certain characteristics about the data, such as normality and independence. If these assumptions are violated, alternative non-parametric tests, which do not rely on specific distributional assumptions, may be more appropriate.
In summary, while the conventional guidelines state that the t-test is used when the sample size is small and the population standard deviation is unknown, and the z-test is used when the sample size is large and the standard deviation is known, there are exceptions to these rules. The choice between the two tests should consider factors such as the sample size, distributional assumptions, and the nature of the research design. Additionally, alternative non-parametric tests can be employed when the assumptions for the z-test or t-test are not met.