[Question] Null and Alternative Hypothesis contradictions

[u/Prestigious-Law65]

My professor is contradictory in her lessons, much of which I'm having to google to figure out what it is she said that's actually true in stats and how much of it is her not gaf.

Right now, I'm having trouble figuring out hypothesis. The questions she has us do through Pearson's AI goes differently, and Pearson isn't always correct either (Pearson is awful and I hate it)

My understanding: (I will do my best with the symbols)

Ho only uses <=, >=, =

Ha only uses <, >, |=

- The claim, according to Pearson, can be either Ho or Ha depending on the equality symbols used.

- But my professor says the null hypothesis (Ho) is ALWAYS the claim and the alternative hypothesis (Ha) is the ALWAYS the complement but applying both rules brings up a lot of contradictory answers depending on the provided data in the questions.

Example: "An Internet provider is trying to gain advertising deals and claims that the mean time the competitors' customers spend online per day is less than 45 minutes. Write the hypothesis."

The claim: u < 45 minutes

My professor's "correct" answer:

Ho: u < 45 minutes

Ha: u >= 45 minutes

- This is contradictory because Ho can't use < and Ha can't use >=.

Pearson's "correct" answer:

Ho: u >= 45 minutes

Ha: u < 45 minutes

- This contradicts my professor saying "Claims HAVE to be the Ho"

Which one is actually correct?

Personally, I think Pearson's is correct and my prof is giving blondes a bad name as usual but going back and forth on her assignments and pearson's "correct" answers is frustrating as all hell and I'm getting discombobulated. I've emailed her many times about previous assignments only to get ghosted or a response that says "Figure it out" or "Ask your adviser" (my advisor teaches biology not stats, how is he supposed to help?) so contacting her about it isn't going to get me anywhere. Unfortunately I am not available when tutoring is open either, so I can't get help there. I used to like math...

[u/seanv507]

OP, nothing is really set in stone. It is important to understand rather than follow rules blindly.

A "claim" has no scientific meaning outside the context (which you have not provided)

eg Cigarette companies claim smoking is harmless. Doctors claim smoking is harmful. Which is the null hypothesis depends on what you are trying to disprove.

Scientifically speaking all you need for the null hypothesis and alternative hypothesis are that they are complements.

Mathematically/Statistically speaking, it is convenient to have the [mathematical] null hypothesis be an equality. Then you extrapolate the scientific question to <= or >= as a conservative underestimation. (ie the alternative result becomes even more unlikely for one side of the inequality, than the equality)

[u/Putraenus_Alivius]

- The claim, according to Pearson, can be either Ho or Ha depending on the equality symbols used.
- But my professor says the null hypothesis (Ho) is ALWAYS the claim and the alternative hypothesis (Ha) is the ALWAYS the complement but applying both rules brings up a lot of contradictory answers depending on the provided data in the questions.

I think what your prof is trying to say is that we always test the null hypothesis and never the alternative hypothesis. We have a claim that differs from an imagined hypothesis that is against that claim, and we test our data against that imagined hypothesis. This imagined hypothesis is the null. We then compare our data to what we should expect to see under the assumption the null is correct, and if it satisfies a certain criteria (usually a p-value inferior to 5%), we reject OR fail to reject the null. At no point do we ever test the alternative hypothesis (the claim we want to prove).

That being said, I do disagree with your professor about the hypotheses. In your example, the claim is the mean time is less than 45 minutes. This means that this should be the alternative hypothesis (HA : u < 45 minutes) whereas the null should be the whenever the claim would be false (i.e. when it's 45 minutes or above so H0 : u >= 45 minutes). We then test that null and we either reject it (giving more credence to the alternative hypothesis) or fail to reject it (giving more credence to the null).

EDIT:

My understanding: (I will do my best with the symbols)
Ho only uses <=, >=, =
Ha only uses <, >, |=

Ehh, this isn't really the case. The HA – the thing we want to prove – can take any symbols. The H0 adapts to that. If I'm testing two drugs for high blood pressure, for example, the standard Drug A and the new Drug B, I can say, 'Drug B lowes systolic/upper blood pressure by at least 20 mmHg compared to the control' and the HA uses a less than or equals to sign. The H0 will just be that the 'less than or equals to -20 mmHg' is 'more than -20 mmHg' because it's the opposite:
H0 : Difference of systolic BP > -20 mmHg
HA : Difference of systolic BP <= -20 mmHg

[u/richard_sympson → u/Putraenus_Alivius]

No, the alternative hypothesis cannot take any symbol. The null hypothesis should specify the parameter lies within a closed set, not an open one, so that you cannot have non-null values for which your test cannot have power.

[u/richard_sympson]

It seems your professor is saying the claim is the thing that needs to be tested—that is, you're testing because you're applying skepticism—and the framework you're using is null hypothesis significance testing, hence the claim is the null hypothesis. This is largely a matter of linguistics and intent, and you are correct that one could just as well switch the direction of your inequalities, and have a new test. But because your professor is the one teaching the class, you may want to just adopt their convention for the course.

That being said, the null hypothesis should specify a closed set, so you are correct that however you interpret the claim WRT null/alternative, the null should use ≤, ≥, or = symbols, and the alternative should use negations of those. If you have

H0 : µ > 0
Ha : µ ≤ 0

then if truly µ = 0, you will never have power to reject the null, even though it is false. Setting the null to contain its closure avoids this issue.

[u/richard_sympson → u/richard_sympson]

Contra to other users too, I am preferential to Fisher's POV on this matter, in that the tools you're likely using (p-values, sampling distributions; unless you're working on LRTs) are indexed to distributions whose parameters lie within the null set. If your null hypothesis is something like H0 : µ ≥ 0, then it is a composite hypothesis, and you take e.g. the p-value to be the supremum of p-values across sampling distributions which comport with the null hypothesis. You generally do not need an alternative hypothesis at all, it has no mechanistic purpose.

[u/jeffcgroves]

Pearson is correct here

my professor saying "Claims HAVE to be the Ho"

A "successful" results rejects the null hypothesis, so your professor is wrong and the NEGATION of your claim has to be the null hypothesis. If you can reject the null hypothesis, you show the negation of your claim is false, meaning your claim is true. It's a double negative, but used correctly

Something important to consider though: you can invert the H0 (consider using zero instead of oh because "Ho" sounds funny) and Ha, and still not get a successful result. In other words, many statistical analysis result in "we can't tell either way".

It's possible your prof was saying that, even if you can't prove "competitors' customers spend less than 45 minutes online per day", you probably can't prove "competitors' customers spend more than 45 minutes online per day" either. The analysis yields an inconclusive result.

In other words "cannot reject the null hypothesis" does NOT mean "accept the null hypothesis". It's possible your prof was somehow alluding to this

[u/rgranger]

Pearson’s is correct. I usually tell my students to start with the alternative hypothesis. The alternative hypothesis is what you’re trying to prove.

[u/Stats_n_PoliSci → u/rgranger]

Just a language note, Ha is what you’re trying to provide evidence for. You can’t prove it with only statistical methods.

[u/seanv507 → u/rgranger]

Yes, but what is the alternative hypothesis?

From the advertiser's point of view, isn't the null hypothesis that the internet provider is "telling the truth"

and the advertiser wants to disprove the null hypothesis? I feel that the question is ambiguous, because it's not quite clear whose point of view we are supposed to be taking.

[u/efrique]

But my professor says the null hypothesis (Ho) is ALWAYS the claim

It depends on the claim (when a claim is actually made), but I'd say in practice it is considerably more often that some claim that is made will end up as the alternative, while the position of withholding acceptance of the claim is the null.

Pearson is awful and I hate it

I have a broadly similar reaction.

[u/Remarkable_Owl_3775]

Estás en lo correcto, la de Pearson está bien, en los juegos de pruebas de hipótesis, la creencia y/o sospecha del investigación es la Ha( ..competencia pasan en linea al día es inferior a 44 min) , pues realmente tu estás probando Ho para pensando en que no sucede nada, un tipo de statu quo y luego poder rechazarla y poder tomar como alterna tu creencia.