If you are trying to decide whether to use a z-test or a t-test, you are trying to determine how well your sample size represents the entire population. A larger sample size will better represent the entire population to a certain point because after a certain size, t and z distributions begin to look the same. If a sample size is to small it will be less representative of the entire population and it is more likely to either miss characteristics of the entire population or over emphasize characteristics. There are different critical values for distributions t and z (Foltz, 2012).
The z-test chart would be best to use with sample sizes over 30 and represents what is considered a normal distribution. If you want to know how many people in a school scored over 95 on a test and you know the average is 85 with a standard deviation of 15, and you take a sample size of 40 people, this would be a good time to use the z-test (Foltz, 2012).
The t-test chart would be best to use with smaller sample sizes and allows greater room for error. The distribution of the t-test curve is more platykurtic when compared to normal distribution or the curve for the z-test. If you wanted to know how many people in a school scored over 95 on a test and you know the average is 85 with a standard deviation of 15, and you take a sample size of 20 people, this would be a good time to use the t-test. The benefit of using the t-distribution is that you can use a smaller sample size with the understanding that there may be a greater margin of error (Foltz, 2012).
It may also be appropriate to use the t-test if the population standard deviation is unknown (Grove & Cipher, 2017).
Z-test and T-test are methods for hypothesis testing (which is the main goal of statistics). Hypothesis testing determines which statements about the population is most reliable based on data available.
For example, a hypothesis that claims those individuals who take Airborne regularly are less likely to become sick during influenza season than those who do not take the supplement. A hypothesis test can be done by giving sample (sample of 40 individuals for instance) doses of Airborne for a certain period of time compared to a control group.
Z-test is utilized for larger set of data or when sample size is greater than 30 (n ≥ 30). Based on the previous example, there were 40 individuals mentioned. Therefore, Z-test is more appropriate to use over T-test. Z-test is done to compare two groups when population variance (standard deviations) are known. Therefore, when standard deviation is known, z-core can be calculated. With normal distribution, researcher can determine probabilities using mean from the sample. This method allows researcher to compare scores from different normal distributions.
Sometimes, samples are small so T-tests are used. T-test is more applicable when participants are less than 30 (n ≤ 30). T-test is applicable when standard deviation is not known and when certain data sets are not suited for analysis using normal distribution. While z-test follows normal distribution, t-test follows t-distribution (or Student’s t-distribution).