Z-tests and t-tests are both commonly used in hypothesis testing, but they serve different purposes and are applied in different scenarios.

Let’s start with the z-test. A z-test is used to compare a sample to a defined population. It determines how far, in terms of standard deviation, a data point is from the mean of a data set. Z-tests are particularly useful for larger sample sizes (n > 30) and when the standard deviation is known. These tests can help identify issues or discrepancies within the studied population. Additionally, z-tests are commonly used when testing a hypothesis.

On the other hand, t-tests are used to determine the statistical difference between two independent sample groups. These tests analyze the likelihood of the observed differences between two samples occurring due to random chance. T-tests are most appropriate when dealing with smaller sample sizes (n < 30). One-sample t-tests, in particular, compare the mean average between a sample and the population being studied. They also examine the difference between the mean of the sample and the mean of the population. The main difference between z-tests and t-tests lies in their assumptions and requirements regarding the data set. Z-tests always assume that the data is normally distributed and necessitate knowledge of the population's standard deviation after data collection. In contrast, t-tests are more flexible and can be used even when the data may not follow a normal distribution. Additionally, t-tests can be applied when the population standard deviation is unknown, making them suitable for situations where information about the population is limited. To summarize, z-tests are used to compare a sample to a defined population, particularly for larger sample sizes and when the standard deviation is known. T-tests, on the other hand, assess the statistical difference between two independent sample groups, often for smaller sample sizes and when the population standard deviation is unknown. Both tests are valuable tools in hypothesis testing, but their specific applications depend on the nature of the data and the research question at hand.