I have two samples where one is a constituent of another, so I have a sample on a stock and a sample on the market index. Should I use the paired t-test or independent t-test to test if the means are equal??
Since this question has been migrated I'll try to keep the jargon to a minimum
First issue for paired analysis is you need one to one correspondence between individual data points in the stock and market sets. So if the individual stock prices are associated with a matched market index then the data is suitable for a paired test.
I'm assuming that the market index you are interested in involves the stock price, so it should be possible to obtain one to one correspondence if sampled correctly since every market index would have a specific value for the stock associated. (if this is wrong please include more detail in the question about what the two samples are)
Now you need to ask what exactly you want to compare. The unpaired test will test if there is a difference between average stock and average market index. The paired test will examine if the difference between stock and market index is on average different from 0.
The paired test is useful when you have an underlying trend that simultaneously affects the two variables as the pairing matches points within this trend it negates its impact. Without this pairing the underlying trends will contribute to the observed variation in both sets and increase the apparent level of noise.
With no underlying shared trend the paired and unpaired tests should give the same result. With a underlying trend the paired test will be more sensitive.
And finally t-test is a test of difference, not equality. While a low p-value that is below your threshold is strong support for a difference existing, a high p-value is not confirmation of equality. The number of ways to get a test to fall vastly exceed the ways to get it to succeed, so if proving equality is what you really need then you need a different test. If that is the case I would recommend framing a new question for the community to help us understand exactly what information you need from the data so we can guide you how to get it