What does a confidence interval that follows a normal distribution require?

A confidence interval that follows a normal distribution requires that the population standard deviation is known. This condition is critical because it impacts the calculation and interpretation of the confidence interval. When the population standard deviation is known, it allows for the use of the Z-distribution to construct the confidence interval, providing accurate estimates of the population parameter.

While it's true that the confidence interval can be constructed under certain conditions when the sample size is less than 30 or when the data is normally distributed, those conditions do not solely define the requirement for a confidence interval to follow a normal distribution. The sample size being less than 30 typically leads to the use of the t-distribution instead of the normal distribution, especially when the population standard deviation is not known. Similarly, for a sample's confidence interval to be valid when the population standard deviation is unknown and the sample size is small, it is often necessary for the data to be normally distributed.

In summary, knowing the population standard deviation is essential for confidently employing the normal distribution and accurately constructing confidence intervals.