So I am a bit confused about the interpretation of the standard error especially when the population standard deviation is unknown. What I've constructed from what I've read about it is that: the mean of the sampling distribution is the population mean and the SD of the sampling distribution is the population standard deviation (sigma) divided by the square root of the sample size. By taking many samples of a reasonable size from a population with a known SD, we can calculate a 95% CI for the mean of the sampling distribution (or is it the mean of the population?). In this case, each sample has the same SE ( which is equal to the SD of the sampling distribution) because we use the same population SD. Is this right?
If the population SD is unknown, we use the sample SD (s) and now each sample has a different SE. By taking many samples of a reasonable size and taking the mean of the means of the sample, we can get an estimate for the mean of the sampling distribution. To calculate the CI, we need the SE which is the sample standard deviation of the sample means divided by square root of the sample size. Is my understanding correct or am I missing something? What exactly do we estimate with the the samples when we don't know sigma and how is the SE interpreted differently from when sigma is known.