Research Idea: the comparison of two long-term variables (2)

In the discussion of t-distribution idea of comparing two long-term variables

Dr. Chen raised a very constructive comment about this idea and suggested that the definition of the long-term exposure is kind of similar to the idea in the survey sampling, where the long-term exposure can be considered as the exact if the everyday measurement is available.

I did the wrong way to interpret the daily measurements within the long-term period to consider its annual mean variance is the variance of annual exposure. If the measurements are complete, the annual exposure should not have any variance in lieu of the mean variance.

The test should be to check if an random variable with mean and variance can be considered as the approximation of the exact value (z-test), where the null hypothesis is to see if the population mean and sample mean are the same.

Once the measurement is incomplete, I should consult with the sampling analysis to see how to use the incomplete sample to have a representative value and its corresponding variance.