Is this null hypothesis wrong?

Question

The Wikipedia article Null hypothesis says in its opening paragraph:

In inferential statistics, the term "null hypothesis" is a general statement or default position that there is no relationship between two measured phenomena, or no association among groups.

(Since this article changes frequently, here is a link to the current version)

The book I'm following has to say :

Q: In a nutritional study 13 students were given a usual diet with vitamin tablets and 12 set of other students were given only the normal diet. After 12 months their weights are measured as given below (a 2 x 13 table in which weight gains of the two set of students are mentioned) Can you say that vitamins were responsible for this difference?

A: $H_0$= Vitamins are responsible for this difference $H_A$= just the opposite, not responsible

The t-value turned out to be more the p-value and they rejected the $H_0$

P.S. I'm sure it isn't right because Vitamin rich diet does cause weight gain and given the concept of null hypothesis the $H_0$ assumed in the excerpt should be wrong.

Answer 1

The idea that a null hypothesis is "a general statement or default position that there is no relationship between two measured phenomena, or no association among groups" assumes that the purpose of the study is to demonstrate that such a relationship or association exists. However that is not always the case, and the null hypothesis ought to be the hypothesis that is to be "nullified" in order to be in a reasonable position to promulgate your alternative (or research hypothesis), i.e. the thing you don't want to be true. Without this, NHSTs do not enforce the self-skepticism that is their main purpose.

For example, climate skeptics often claim that there has been a pause in the underlying rate of global warming (as measured by global mean surface temperatures - GMSTs). However the evidence they use for this is a lack of a statistically significant trend since some (usually cherry picked to coincide with El-Nino) starting point. However in doing so, they are using a null hypothesis that there is no warming, which is also the alternative hypothesis for which they are arguing. This totally eliminates the value of NHSTs in making us question the evidential support from the data. Of course what they should do is assume a null hypothesis that states that warming has continued at the same rate predicted by the climate models and try and see if that H0 can be rejected (it can't at the moment, as far as I am aware).

Similarly, in the case of the vitamin study, it depends what the authors were trying to argue. If they were arguing that vitamins do cause, then a H0 corresponding to "no effect" would be more appropriate, however I think the real problem in this case is more likely to be limited power of the test? "Vitamin rich diet does cause weight gain" I don't think this is true, if you had a diet of only vitamin pills, I'm pretty sure you would loose weight quite rapidly.

Incidentally, a paper worth reading is

Gerd Gigerenzer, Stefan Krauss, and Oliver Vitouch, "The Null Ritual What You Always Wanted to Know About Significance Testing but Were Afraid to Ask", in D. Kaplan (Ed.). (2004), The Sage handbook of quantitative methodology for the social sciences, (pp. 391–408). (pdf)

An element of the "null ritual" that they are criticizing is:

1) Set up a statistical null hypothesis of “no mean difference” or “zero correlation.” Don’t specify the predictions of your research hypothesis or of any alternative substantive hypotheses.

So H0 is not automatically the assumption of "no effect/difference/correlation/association", it's definition depends on your research definition. If you don't say what you are trying to argue for you are in no position to state your null hypothesis.

Answer 2

The book's stated null is not a null -- it's not even suitable for a one-sided null. The part in the Wikipedia definition you quote should be sufficient to guide you to putting the hypothesis listed under the alternative in for the null (but hopefully properly rephrased into a more precise statement about a population)

However, that main issue of your question aside, the definition of a null hypothesis that you quote from Wikipedia is itself not remotely sufficient as a definition of a null hypothesis, since it omits valid nulls. One could excuse a definition in an opening sentence that omitted some odd edge-cases, but it excludes a host of quite common "intro-stats text-book" null hypotheses.

Consider a null about a parameter value for a single population (like $H_0: \mu=100$) -- that's not covered by that purportedly general statement telling us what a null hypothesis is. Or consider a null hypothesis in a goodness of fit test, like say a Kolmogorov-Smirnov test -- that's also not covered by that definition of what a null hypothesis is.

So while it should have been sufficient to guide you to the correct conclusion here (that the answer is flat out wrong), you should beware trying to use it as in any sense a definition of a null hypothesis, in spite of it being exactly what the opening sentence of the article implies.

(I'd also add that even if it gave a reasonable definition that didn't exclude a host of standard cases - admittedly a tricky task if you want a concise statement - that opening sentence doesn't even correctly say what it means to say; it conflates the term with the thing the term is a convenient name for. The article says 'the term "null hypothesis" is a general statement...'. No, a null hypothesis may be a kind of "general statement" about something, while the term null hypothesis is what that thing is usually called. The term isn't actually the thing -- we should not expect the term 'dog' to eat dog-biscuits and play fetch; indeed a few quick experiments suggest that this three-letter term is singularly unsuccessful at munching doggie treats and soaking balls in drool; even a fairly recalcitrant cat performed better than the term 'dog' did, since at least it displayed some passing interest in the dog-biscuit. The phrasing there brought to mind the discussion of the White Knight's song in Through the Looking Glass)

Answer 3

I think you should use as your null hypothesis Ho: Vitamins don't have any effect at all on the weights, vs Ha: Vitamins do have an effect on the weights.

This is for a bilateral hypothesis testing actually, if you wanted to do an unilateral test, the alternative hypothesis should be Ha: Vitamins affect increasing the weight (or Vitamins affect decreasing the weight).