If we want to test whether the sample mean is at least 5 meters, how should we state the null hypothesis and alternative hypothesis? Here is what I think: H0: µ>=5 H1: µ<5 But when determining the p value, I will be looking at a left tail probability isn't it? It just feels weird. So is my H0 and H1 correct?
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2 Answers
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You do not want to take H0 as µ>5 because it can't yield assumptions that allows you to draw a distribution to test it. This is the point of null hypothesis testing. Only H0:µ=5 allows you to do that. Moreover, in null hypothesis testing, you want to reject H0 (not accept it), then in any case µ>5 was the wrong way to go. More precisely, I would advocate for :
- H0 : µ=5
- H1 : µ>5
- You look at the right tail of your z-distribution under H0 (in order to reject H0 to the benefit of H1).
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Using the usual test statistic based on $H_0\colon\{\mu = 5\}$ and setting either $H_0\colon\{\mu \geq 5\}$ or $H_0\colon\{\mu = 5\}$ does not change anything to the significance level of test, defined as $$\max_{\mu \in H_0}\,\Pr(\text{reject } H_0 \mid \mu),$$ because this $\max$ is attained for $\mu=5$. This is expected: if you have evidence that $\mu \geq 5$ is not true then a fortiori you have evidence that $\mu \geq 6$ is not true.
About the $p$-value, I don't know whether there is a consensual definition allowing for the case of non-"sharp" hypotheses. But we could use a $\max$ too.
answered May 19, 2015 at 8:55
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$\begingroup$ Please downvoter, could you explain why my answer deserves your downvote ? $\endgroup$ May 19, 2015 at 15:54
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$\begingroup$ Maybe your answer doesn't address the question? $\endgroup$ May 20, 2015 at 0:56
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$\begingroup$ @StudentT My answer addresses the question how should we state the null hypothesis. $\endgroup$ May 20, 2015 at 5:33
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$\begingroup$ But you weren't directly addressing like brumar did in the answer. $\endgroup$ May 21, 2015 at 5:35