Videos
I teach a probability and statistics course in a university but I'm teaching outside my field so I'm definitely not an expert. I have a question about choosing the null and alternative hypotheses and haven't been able to resolve it via googling. I teach in an engineering department so examples about drug testing aren't as relevant.
Question: does the choice of Ho and Ha depend on which "side" of the claim you're on, ie if you want to prove or disprove it?
Let's say a lightbulb manufacturer claims their bulbs last on average at least 800 hours. If I work for the manufacturer, I want to conclusively demonstrate via my hypothesis test that my claim is true, so it seems that I would want Ho : mu <= 800 and Ha : mu > 800 so that I could reject Ho with a certain level of significance and be confident in my claim.
However if I'm a consumer and I don't believe the manufacturer's claim, it seems that I want Ho and Ha to be the reverse, so I could conclusively determine that their claim is false and that the true lifespan is less than 800 hours, so that I'd have evidence that they're being dishonest.
Can anyone confirm if the above logic is correct, that sometimes the choice of whether the stated claim is Ho or Ha depends on if you want to prove or disprove the claim?
Thanks in advance!
Edit: here's an example from the textbook, for an idea of the types of problems I'd like to be able to write:
A manufacturer of a certain brand of rice cereal claims that the average saturated fat content does not exceed 1.5 grams per serving. State the null and alternative hypotheses to be used in testing this claim and determine where the critical region is located.
Solution: The manufacturer’s claim should be rejected only if μ is greater than 1.5 milligrams and should not be rejected if μ is less than or equal to 1.5 milligrams. We test
H0: μ = 1.5,
H1: μ > 1.5.
Nonrejection of H0 does not rule out values less than 1.5 milligrams. Since we have a one-tailed test, the greater than symbol indicates that the critical region lies entirely in the right tail of the distribution of our test statistic Xbar.
To me, this problem seems to be written from the perspective of a test engineer at the FDA who wants to try and prove the company's claim wrong. If I worked for this manufacturer, wouldn't I want to switch H0 and H1, so that I can reject the claim that mu>1.5?
On principle, there is no reason for hypotheses to be exhaustive. If the test is about a parameter $\theta$ with $H_0$ being the restriction $\theta\in\Theta_0$, the alternative $H_a$ can be of any form $\theta\in\Theta_a$ as long as $$\Theta_0\cap\Theta_a=\emptyset.$$
An example as to why exhaustivity does not make much sense is when comparing two families of models, $H_0:\ x\sim f_0(x|\theta_0)$ versus $H_a:\ x\sim f_1(x|\theta_1)$. In such a case, exhaustivity is impossible, as the alternative would then have to cover all possible probability models.
The main reason you see the requirement that hypotheses be exhaustive is the problem of what happens if the true parameter value is in the region which is not covered by either the null or alternative hypothesis. Then, testing at the $\alpha %$ level of confidence becomes meaningless, or, perhaps worse, your test will be biased in favor of the null - e.g., a one-sided test of the form $\theta = 0$ vs. $\theta > 0$, when actually $\theta < 0$.
An example: a one-sided test for $\mu = 0$ vs $\mu > 0$ from a Normal distribution with known $\sigma = 1$ and true $\mu = -0.1$. With a sample size of 100, a 95% test would reject if $\bar{x} > 0.1645$, but 0.1645 is actually 2.645 standard deviations above the true mean, leading to an actual test level of about 99.6%.
Also, you rule out the possibility of being surprised, and learning something interesting.
However, one can also look at it as defining the parameter space to be a subset of what might typically be considered the parameter space, e.g., the mean of a Normal distribution is often considered to lie somewhere on the real line, but if we do a one-sided test, we are, in effect, defining the parameter space to be the part of the line covered by the null and alternative.