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?
