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I'm wondering what role t-statistic plays in hypothesis testing and the interpretation of the result.

In a two-sample two-tailed t-test where we have $H_0$: $\mu_A=\mu_B$ and $H_1$: $\mu_A \neq \mu_B$, I'm wondering the difference between the interpretation of the following two cases.

  • Case 1: $p<\alpha$ and $t<0$
  • Case 2: $p<\alpha$ and $t>0$

Given that $p<\alpha$, we reject $H_0$ and accept $H_1$ that $\mu_A \neq\mu_B$ in both cases, but how does the positivity or negativity of $t$ influence our interpretations of the two cases?

I have the same question for the one-tailed t test. Say that now we have $H_0$: $\mu_A\leq\mu_B$ and $H_1$: $\mu_A > \mu_B$ and we also consider the following two cases:

  • Case 1: $p/2<\alpha$ and $t<0$
  • Case 2: $p/2<\alpha$ and $t>0$

Then, how does the positivity/negativity of $t$ influence the two cases above in the one-tailed t-test?

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  • $\begingroup$ In method 2, have you not mixed up the inequalities? $\endgroup$ Jan 28, 2022 at 18:51
  • $\begingroup$ @RichardHardy Sorry I'm not entirely sure what you meant. Or is it actually $H_0$: $\mu_A\leq\mu_B$ and $H_1$: $\mu_A>\mu_B$? I'm actually not sure which is the case but I'm sure that some people use inequalities in the null hypothesis, which confused me about whether we should use an equality or inequality in the null hypothesis. $\endgroup$
    – Jack
    Jan 28, 2022 at 19:03
  • $\begingroup$ You might find this relevant. $\endgroup$
    – dimitriy
    Jan 28, 2022 at 19:13
  • $\begingroup$ @dimitriy Thanks a lot! I think this is useful! It mentioned that $H_0$ is the hypothesis that we want to argue against. Based on that, I think $H_0$ in Method 2 should be $\mu_A\leq\mu_B$. I'll make changes in my post accordingly. $\endgroup$
    – Jack
    Jan 28, 2022 at 19:19
  • $\begingroup$ Also, $H_0$ and $H_1$ should be mutually exclusive, so Method 1 is not correct and its $H_0$ should be corrected as $\mu_A\leq\mu_B$, which is just the same as Method 2. The article just provides the correct way to define the two hypotheses (assuming the article is correct). $\endgroup$
    – Jack
    Jan 28, 2022 at 19:23

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