I have a sample with a sample size of > 50,000 (With a mean of 0 and a known standard deviation).
Now, I'm picking a subsample of the bigger sample with a sample size of 5,000 (With a mean of 2 and a known standard deviation).
The goal is to find out, if the mean of the subsample is significantly differnt to zero (not different to the mean of the bigger sample, but to zero!). Therefore, I would do a one-sample t-test (wikipedia one-sample t-test).
Which standard deviation s is the right one to use? Following wikipedia, the standard deviation of the subsample should be taken. However isn't this usually done, because the standard deviation of the subsample is usually the best approximation of the standard deviation of the whole population? Wouldn't it be more accurate to take the standard deviation of the bigger sample in my case?
Edit for clarification (Hope this helps):
I have the following:
- Whole Population: Mean is known (=0), Standard Deviation (SD) is unknown
- Subpopulation (Sample): Mean is known (=0), SD is known (representative for whole population)
- 10 Subsamples of subpopulation: Mean is known ( !=0), SD is known and can be slightly different to subpopulation SD.
The subsamples are created by a classification, which was done before. But, the subamples are not classified by the variable, which is now used for calculating the mean or SD! In my understanding, the subsamples are therefore random.
Imagine the following simplified example:
- Whole Population: Stock universe
- Subpopulation: Representative Index
- 10 subsamples: Stocks ranked by their market capitalization (e.g. subsample one consists of all the smallest stocks, subsample 10 of the biggest stocks)
- Target Variable: Relative returns of the stocks to the index. Mean and SD are calculated over the returns of all stocks in one subsample.
Information which I want from the t-test:
- Is the mean of a subsample significantly different to zero
- Is the t-test the right way to do this?
- If yes, do I use the subsample SD or the subpopulation SD for calculating the t-values?