Confidence interval given the population mean and standard deviation Say we have a normal distribution with mean 10 and standard deviation 5. How do I get a two sided and one sided (left/right) 90% confidence interval using this?
This is the actual question I am trying to answer:
We have a normal process with mean 10 and standard deviation 5. 

We want to test whether or not the values 15, 12, and 4 come from this process.

Are those three values in a 90% symmetric about the mean confidence interval?
Are those three values in a 90% one-sided to the left confidence interval?

Everything about confidence intervals I see online refers to cases where the population mean is unknown, and you create a confidence interval using the sample mean, sample standard deviation, and sample size. I can't find anything when you have the population mean and standard deviation and then want a confidence interval for the data.

If you have the population mean, and the population variance, you would not calculate a confidence interval for the mean of a normal distribution.
Here's why: Confidence intervals are one form of statistical inference. The aim of statistical inference is to learn something about some statistical quantity or quantities in a target population, given that what what we have to learn from are data and statistical quantities derived with them from a sample.* Loosely, confidence intervals are intended to provide a plausible range of estimates of the statistical quantity in the target population, where this range gets wider as our level of confidence increases (e.g., ceteris paribus a 99%CI is wider than a 90% CI). If we already know the statistical quantity in the population, we have no need to try and infer it. Finally, everything about the normal distribution is conveyed through it's two parameters, the population mean $\mu$ and the population standard deviation $\sigma$ (though some prefer to think of the population variance, $\sigma^{2}$ as the parameter): if you know these quantities in advance, you do not need to guess them, as @Dave said.
* I am intentionally leaving out additional things we have to work with like modeling assumptions, prior beliefs, and so forth.
