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In Chapter 8, Test of Hypotheses based on a Single Sample, in Devore's Probability & Statistics for Engineering and Sciences, he states that the null hypothesis, $H_{0}$, is the a priori claim (the claim believed to be true), while the alternate hypothesis, $H_{a}$, is the assertion contradictory to $H_{0}$.

He goes on to state that:

Sometimes an investigator does not want to accept a particular assertion unless and until data can provide strong support for the assertion.

And gives this example:

Suppose a company is considering putting a new type of coating on bearings that it produces. The true average wear life with the current coating is known to be 1000 hours. With $\mu$ denoting the true average life for the new coating, the company would not want to make a change unless evidence strongly suggested that $\mu$ exceeds 1000. An appropriate problem formulation would involve testing $H_{0}: \mu = 1000$ against $H_{a}$: $\mu > 1000$.

I get this. There must be compelling evidence that the new coating's wear life is >= 1000 hours. Unless that evidence is provided, we must accept that the best wear life for a coating is equal to 1000 hours.

However, Question #23 in Section 8.2, is the following:

Exercise 36 in Chapter 1 gave $n = 26$ observations on escape time (sec) for oil workers in a simulated exercise, from which the sample mean and sample standard deviation are 370.69 and 24.36, respectively. Suppose the investigators had believed a priori that true average escape time would be at most 6 min (360 secs). Does the data contradict this prior belief? Assuming normality, test the appropriate hypotheses using a significance level of $.05$.

I wrote my null hypothesis and alternate hypothesis as the following: $$ H_{0}: \mu = 360 $$ $$ H_{a}: \mu < 360 $$

My reasoning being that the burden of proof should be on proving that the escape time is less than 6 minutes. We must assume that the escape time is 6 minutes (or even more). In this case, it seems like I want to err on the side of making a Type II error. I want to fail to reject the null hypothesis even when it's false (i.e. I want to assume the worst-case scenario).

However, the student solutions manual provides null and alternative hypothesis as:

$$ H_{0}: \mu = 360 $$ $$ H_{a}: \mu > 360 $$

Isn't this like saying that the burden of proof is in proving that the escape time is greater than 360 seconds? In this case, I want to err on the side of making a Type I error. I want to reject the null hypothesis of 360 seconds, even if it is true, and err on the side of caution.


So people, where is the error in my logic? If my alternate hypothesis is indeed $H_{a}: \mu > 360$, then I better make sure that the burden of proof isn't too great... right?

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    $\begingroup$ One the information presented, I see no solid basis on which to do a one-tailed test there at all. Perhaps there was information in the earlier exercise that helps there but otherwise I don't see why the one tailed test was used in this case. $\endgroup$
    – Glen_b
    Apr 27 '15 at 0:55
  • $\begingroup$ There are two possible outcomes in hypothesis testing: a) Reject the null hypothesis in favor of the alternative, and b) Fail to reject the null hypothesis. If as you proposed, $H_0$ is $\mu=360$ and $\mu<360$, you have no evidence for outcome a), because a sample with mean $>360$ cannot cause you to believe the chosen alternative, which is that the actual mean may be $<360$. You found a sample mean $>360$, so how can that cause you to believe the actual mean is on the other side of 360? $\endgroup$
    – Steve Kass
    Apr 27 '15 at 3:59
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I think the problem might be that these questions really ought to be thought of as 2-tailed. The alternative is not that the new coating is better than the old coating, but that the new coating is different: it might be better or worse. This is true even if the company is only interested in using the new coating if it is better. If you design the study for a one-tailed analysis, the sample size will call for many fewer samples than for two-tailed. However, nature is cruel and indifferent to your desires to only find better coatings - the coating might be worse and you had better plan your analysis accordingly. For this reason, there are very few situations where a one-tailed test is truly indicated. I think the null hypothesis ought to be Ha> or < Ho.

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I believe the answer isn't from looking at Type I and II errors. It revolves around "rejecting the null hypothesis" and "failure to reject the null hypothesis."

"Suppose the investigators had believed a priori that true average escape time would be at most 6 min (360 secs)." Therefore setting up the hypothesis as

$$ H_{0}: \mu = 360 $$ $$ H_{a}: \mu > 360 $$

makes our prior belief to be $H_{0}$, and we are trying to "reject the null hypothesis" and see if our evidence can reject it. If we setup the other way

$$ H_{0}: \mu = 360 $$ $$ H_{a}: \mu < 360 $$

gives an alternative hypothesis that is the prior belief stated. We never "accept the alternative hypothesis" and only "fail to reject the null hypothesis." However, since we are trying to see if our data is enough evidence against our prior belief, we are now trying to reject "reject the alternative hypothesis." It also doesn't make sense to "reject the null hypothesis" in this case since it is somehow saying "reject the evidence" and seeing if our "evidence rejects the evidence"? Here are a few links to other (much better) explanations of how to select $H_{0}$ and $H_{a}$

https://people.richland.edu/james/lecture/m170/ch09-int.html

http://faculty.washington.edu/bare/qs381/hypoth.html

https://people.richland.edu/james/lecture/m170/ch09-int.html

https://liesandstats.wordpress.com/2008/09/08/accept-the-null-hypothesis-or-fail-to-reject-it/

http://blog.minitab.com/blog/understanding-statistics/things-statisticians-say-failure-to-reject-the-null-hypothesis

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