Students learning statistics are always troubled with this question: should I use the t-test (t distribution) or the z-test (z distribution)? Every MBA student encounters statistics or data analysis in their core classes and faces these questions.
- Which test should I use? Should I use the z-test or the t-test?
- Which distribution can I use: the z distribution or the t distribution?
- Can I use the t-test or the z test if the sample size is smaller than 30?
- Does it matter if I am working on with sample means vs. sample proportions?
- Is it mandatory to use the z test if the sample size is more than 30 and is considered a large sample?
- Can I use the z test if the sample size is large, but I don’t know the population standard deviation?
As statistics tutors, we encounter these questions frequently!
Answer: Use the Z test if you know the population standard deviation.
You should use the z test and z distribution if you know the population standard deviation. The answer to the question: when can I use the t-test or z test should be as clear as this simple statement. You should use the z test and z distribution if you know the population standard deviation.
Use the t-test if you do not know the population standard deviation.
A corollary to the above rule: Use the t-test if you do not know the population standard deviation and must estimate it using the sample standard deviation.
Defaulting to the t-test. When in doubt, use the t-test!
Why is my professor suggesting that we default to the t distribution? We rarely know the actual population standard deviation. And when we do not know the actual population standard deviation, we should use the t distribution. This rule applies even when the sample size is larger than 30. So, you are rarely wrong if you use the t distribution.
However, as statistics practitioners in the real world, you must remember that the t distribution is a little flatter and has thicker tails. So, while the t distribution is more forgiving than the z distribution, your confidence intervals are usually wider and may be less valuable.
Exceptions to the Z Test vs. T Test Decision Rule: 1) Distribution of the underlying population.
Now some exceptions can be allowed in specific conditions. For example, the z test can be used instead of the t test even if the true population standard deviation is not known if you know that the underlying population is normally distributed and you have a sample size larger than 30.
Exceptions to the Z Test vs. T Test Decision Rule: 2) Large Sample size.
While not technically correct, some faculty allow students to use the t test or z test if they are working with large sample sizes. A large sample size is a sample size greater than 30. The central limit theorem states that the distribution of sample means becomes closer and closer to a normal distribution as the sample size gets larger, regardless of the underlying population’s distribution. So check with your statistics guide or faculty what the expectation is.
What is the difference between the Z Test and T Test? Or Z Distribution vs. T Distribution?
If it is so simple, why is there all this confusion on which test to use: The z test or t test? This confusion may be because the t distribution and z distribution are very similar.
Both the z distribution and t distribution have similar-looking curves. The area under both curves is 100%! The z distribution is higher in the middle. The t distribution has fatter tails. However, as the sample size gets larger (degrees of freedom) gets larger, the t distribution gets closer and closer to the z distribution. It should exactly match the z distribution at infinity! Even at a sample size of 30 or more, the t distribution is very close to the z distribution. It is so close many professors allow you to use the z distribution instead of the t distribution as an approximation!
Therein lies the confusion for many students. There should not be any confusion on When you use the t-test and t distribution) vs. z-test and z distribution? We hope this page gives you the clarity you need to pick the right distribution or test to use. Now go use the t-test or z-test appropriately chosen. Just enjoy learning!