I have a question regarding hypothesis testing. I used the t-test (2-tailed) for these hypotheses:

1. Whether the (monthly) mean return of company A's stock is different from 0
2. Whether the (monthly) mean return of S&P500 is different from 0
3. Whether the (monthly) mean return of company A's stock is different from (monthly) mean return of S&P500.

The result of my tests showed that the mean return of company A's stock is 0, while rejecting the hypothesis the the mean return of S&P500 is 0. However, the result of the last test showed that the mean return of the company A's stock is the same as that of S&P500.

Can someone explain what happened here? And what is the reason behind that?

Thank you so much for your help.

• In the third one, the sample variance used for the denominator of the test-statistic is probably pooled so that might be the cause of the difference. In fact, even if it's not pooled, the expression for the sample variance in 3) is still quite different than the corresponding expressions for 1) and 2) so what you are seeing could be due to that. Jul 19, 2023 at 17:23
• It might help if you can you give the number of observations, the means and the variances in the various cases. Jul 19, 2023 at 19:24

Consider flipping 2 coins and getting 9 and 6 heads on each. You will conclude (say 90% confidence level) that first coin is biased, second is unbiased but the hypothesis both have the same bias cannot be rejected. Obviously this is a very likely outcome if the joint bias is $$Pr(head)$$ in $$[0.7,0.8]$$.