Why Student's T-test? (Part 1)

Here I am trying to answer two questions for myself:

1. What is the difference between Z-test and T-test?
2. Why we need student's T-test?

First, let's be clear on Z-test V.S. T-test. A thumb rule can be referred as, Z-test is used when the sample size is more than 30 while T-test is used for smaple size less than 30. Now let's get back to the history of story:

Sometimes, measuring every single piece of item is just not practical. That is why we developed and use statistical methods to solve problems. The most practical way to do it is to measure just a sample of the population. Some methods test hypothesis by comparison. The two of the more known statistical hypothesis tests are the T-test and the Z-test. Let's try to break down the two.

Strictly speaking, the Z-test is a test for populations rather than samples. In the real world though, either test will give you a pretty close answer. using the T-test is more accurate because the sample deviation is specific and tailored to the sample you are studying, so the answer will be more accurate. When using a T-test of significance, it is assumed that the observations come from a population which follows a Normal distribution. This is often true for data that is influenced by random fluctuations in environmental conditions or random measurement errors. Whereas the T-distribution is essentially a corrected version of the normal distribution in which the population variance is unknown and hence is estimated by the sample standard deviation.

There are various T-tests and two most commonly applied tests are the one-sample and paired-sample T-tests. One-sample T-tests are used to compare a sample mean with the known population mean. Two-sample T-tests, the other hand, are used to compare either independent samples or dependent samples.

As mentioned above, T-test is best applied, at least in theory, if you have a limited sample size (n < 30) as long as the variables are approximately normally distributed and the variation of values in the two groups is not reliably different. It is also great if you do not know the populations’ standard deviation. If the standard deviation is known, then, it would be best to use another type of statistical test, the Z-test. The Z-test is also applied to compare sample and population means to know if there’s a significant difference between them. Z-tests always use normal distribution and also ideally applied if the standard deviation is known. Z-tests are often applied if the certain conditions are met; otherwise, other statistical tests like T-tests are applied in substitute. Z-tests are often applied in large samples (n > 30). When T-test is used in large samples, the T-test becomes very similar to the Z-test. There are fluctuations that may occur in T-tests sample variances that do not exist in Z-tests. Because of this, there are differences in both test results.

Summary:


1. Z-test is a statistical hypothesis test that follows a normal distribution while T-test follows a Student’s T-distribution.
2. A T-test is appropriate when you are handling small samples (n < 30) while a Z-test is appropriate when you are handling moderate to large samples (n > 30).
3. T-test is more adaptable than Z-test since Z-test will often require certain conditions to be reliable. Additionally, T-test has many methods that will suit any need.
4. T-tests are more commonly used than Z-tests.
5. Z-tests are preferred than T-tests when population standard deviations are known.