# Comment 1 Z-tests are used to compare the collected data of a defined population and the sample.  The z-score tells you how far, in standard deviation, a data point is from the mean or average of a data set. A z-test compares a sample to a defined population. Normally, a z-test identifies issues with larger samples that are being studied (n > 30). Z-tests are also useful when testing out a hypothesis. Generally, they are most useful when the standard deviation is known. T-tests are calculations used to test a hypothesis. However, t-tests are used to determine the statistical difference between two independent sample groups. In other words, a t-test analyzes how likely the difference between two samples occurs due to random chance. Usually, t-tests are most appropriate to use when dealing with problems with a limited or small sample size (n < 30). One-sample T-tests are used to compare the mean average between the sample and the population studied. The test is also applied compare the difference between the mean of the sample and the mean of the population. Z-tests always use normal distribution and should be used when the standard deviation is known after the data is collected. Comment 2. A z-score and a t- score are both used in hypothesis testing. Generally, in elementary stats and AP stats, you’ll use a z-score in testing more often than a t score. Z-scores are a conversion of individual scores into a standard form. The conversion is based on your knowledge about the population’s standard deviation and mean. A z-score tells you how many standard deviations from the mean your result is. You can use your knowledge of normal distributions (like the 68 95 and 99.7 rule) or the z-table to determine what percentage of the population will fall below or above your result.  Like z-scores, t-scores are also a conversion of individual scores into a standard form. However, t-scores are used when you don’t know the population standard deviation. Population normal and variance known (for any sample size) 1. Population normal, variance unknown and   n > 30 n>30 (due to CLT) 2. Population binomial,  n p > 10 np>10,   n q >10 nq>10 1. Population normal, variance unknown and   n < 30 n<30 2. No knowledge about population or variance and   n < 30 n<30, but sample data looks normal / passes tests etc. so population can be assumed normal.

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.