# What is the inferential role of a point alternative hypothesis used for planning?

Say I plan an experiment to test the null hypothesis of $$H_0:\mu_1-\mu_2 = 0$$ and, with a power analysis for a Student's $$t$$-test for independent samples, I find that samples of $$n=10$$ (per group) will give me power of 99.5% with the alternative hypothesis of $$H_a: \mu_1-\mu_2=3$$ with $$\alpha=0.005$$.

(Assume the statistical model is a good enough match to reality for the purposes of the analysis.)

Now I run the planned experiment and reject the null hypothesis. What is my inference regarding $$H_a$$?

It seems to me that any time I reject $$H_0$$ I implicitly accept the negation of $$H_0$$, but should I accept the planned point value of $$H_a$$?

A test where I accept the planning alternative whenever I reject the null would allow me to accept an alternative when the evidence do not actually favour it.

(Yes, there are several other questions an answers that are similar, but I have not found any that is explicit on this point, and I think clarity here will be useful.)

It seems you understand it just fine: evidence against null cannot be immediately translated into evidence for any particular $$H_a$$ (unless $$H_a$$ is just the negation of $$H_0$$). For an obvious example, a significant p-value from a two-tailed t-test tells you nothing about the effect direction.
If you need inference beyond null hypothesis testing, a small step forward is looking into confidence intervals, but the final answer is Bayesian analysis: then your posterior support for $$H_a$$ will be determined from prior support for $$H_a$$ and new evidence for it. In practice, this is often replaced with a qualitative discussion of probable effect sizes, power, and evidence ("we had good power to detect any effects larger than $$c$$..."). IMHO, this requires just as much effort and provides less answers than the Bayesian framework.