# Is this a valid argument why null hypothesis should always be equality?

I took an intro stats course in college, I vaguely remember my TA explained why null hypothesis should always be equality, something like the following:

by default we can only assume there’s no change, if we already assume treatment has certain effect on the sample, then we are having a biased assumption for null hypothesis, thus, defeating the purpose of having a null hypothesis. After all, it’s called null hypothesis.

Is this a valid argument why null hypothesis should always be equality?

I was taught in college null hypothesis should always be equality, now I found out it's not true, null hypothesis can be inequality. How come textbooks/instructors still hold on to teaching null hypothesis as equality (especially for one-sided test)? Does null hypothesis stated as inequality has any effect on increasing Type I error?

• Null hypothesis can be inequality. But it must be either $\leq$ or $\geq.$ This is because of calculation pf $p$-value. When do we calculation, if null hypothesis has no equality, I think calculation cannot be made and thus $p$-value cannot be calculated? – Idonknow Dec 24 '19 at 12:33
• In addition to the other answers: The null hypothesis doesn't have to state that there is "no effect" or "no difference". The null hypothesis $\mu_{A}-\mu_{B} = 5$, for example, is perfectly valid. – COOLSerdash Dec 24 '19 at 12:57

## 1 Answer

No, that isn't valid.

For instance, suppose you are interested in the relative heights of adult men and women in different parts of the world. You could find the usual difference across a broad population (say it's 9 cm, just for a guess). Then you look at (say) a tribe in New Guinea. Your null could be "Men in New Guinea are 9 cm taller than women".

The reason many introductory books say that the null is equality is that a) It usually is and b) That's simpler and statistics is hard enough.

• Although correct and revealing, this answer doesn't address the question because its null still concerns an equality (formally, $H_0:\mu_1 - \mu_2 = 9$). The question asks about inequalities. This (strongly) suggests a good answer ought to invoke the distinction between simple and composite hypotheses. – whuber Dec 24 '19 at 16:09
• @whuber Seems to me that Peter Flom's answer addresses the howler that's in the textbook well enough, and that composite nulls are not the issue behind the question. The part "then we are having a biassed assumption" indicates a very bad model of how and what statistics should be used for. – Michael Lew Dec 24 '19 at 20:40