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I am still confused about Hypothesis testing. How does one set up the null hypothesis $H_0$ and the alternative hypothesis $H_a$? I have read a post here that doesn't give it much credit, as far as I can tell. (One must use both $H_0$ and $H_a$ for the AP Statistics exam.)

I have read that $H_a$ is to be what one does not want to be false (to happen). Also, one source says that $H_0$ should always have the = sign, even when doing a one-sided (-tailed) test. But another uses the $\geq$ and $\leq$ in $H_0$ when doing a one-sided test. Another doesn't even mention $H_a$ and just goes straight to the P value.

Which is correct?

Question: Is A bigger than B?

$$H_0:= \left(\mu_a-\mu_b\right) = 0\quad \text{or}\quad H_0:= \left(\mu_a-\mu_b\right) \geq 0$$

$$H_a:= \left(\mu_a-\mu_b\right) \lt 0\quad \text{or}\quad H_a:= \left(\mu_a-\mu_b\right) \lt 0$$

Or is this setup better?

$$H_0:= \left(\mu_b - \mu_a\right) = 0\quad \text{or}\quad H_0:= \left(\mu_b-\mu_a\right) \leq 0 $$

$$H_a:= \left(\mu_b-\mu_a\right) \gt 0\quad \text{or}\quad H_a:= \left(\mu_b-\mu_a\right) \gt 0$$

Peter Flom
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MaoYiyi
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    Every one of those hypotheses is wrong. Hypotheses are about population parameters, not sample quantities. – Glen_b Dec 13 '12 at 09:16
  • @Glen_b How can I fix this? – MaoYiyi Dec 13 '12 at 14:10
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    For some reason, the Internet has an amazing amount of misrepresentations of hypothesis tests. Over a decade ago I collected a few of these and offered criticism. (It was perhaps a little harsh--every one of the statements I quoted had a grain of truth to it--but somehow I am not surprised that every one of those links is now a 404, while most of the links I found useful are still live.) That criticism may help resolve parts of the present question--but no doubt will lead to further questions :-). – whuber Dec 13 '12 at 15:41
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    Shouldn't the proposed $H_0$ be an equality not an inequality? – Jase Dec 13 '12 at 16:07
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    @whuber thank you very much, What you wrote is better than what I read. I really hope you are a teacher because I would like to take your classes. – MaoYiyi Dec 13 '12 at 16:08
  • Mao, @Jase makes a good point: your alternative hypotheses and null hypotheses overlap in the first half of each disjunctive pair. Perhaps you intended to write "=" where you currently have "$\neq$? (BTW, thank you for your kind words in the previous comment.) – whuber Dec 13 '12 at 16:20
  • @MaoYiyi Your hypotheses need to be about $\mu$'s (or $\theta$'s or somesuch symbols for the population means). You already know the $\bar{x}$'s differ! – Glen_b Dec 13 '12 at 21:57

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