When to Use the TDistribution vs. the Normal Distribution for Confidence Interval and Hypothesis Testing Problems for Means
Main Point to Remember:
You must use the tdistribution table when working problems when the population standard deviation (σ) is not known and the sample size is small (n<30).
General Correct Rule:
If σ is not known, then using tdistribution is correct. If σ is known, then using the normal distribution is correct.
What is Most Common Practice:
Since people often prefer to use the normal, and since the tdistribution becomes equivalent to the normal when the number of cases becomes large, common practice often is:

If σ known, then use normal.

If σ not known:
What is Another Common Way Textbooks Teach This:
Textbooks often simplify this to “largesample” vs. “smallsample” methods; use normal distribution with large samples and tdistribution with small samples. This is right almost all the time, because in real sampling problems we seldom have a basis for knowing σ. However, there can be some situations when we do have a basis for assuming a value for σ, such as using a σ based on past data, and in those situations even if sample size is small the correct procedure would be to use the normal distribution, so the simplified “largesample” vs. “small sample” approach would lead to an error. 