# Quants: what is the rationale behind framing of the Null Hypothesis here?

A bottler of iced tea wishes to ensure that an average of 16 ounces of tea is in each bottle. In order to analyze the accuracy of the bottling process, a random sample of 150 bottles is taken. Using a t-distributed test statistic of -1.09 and a 5% level of significance, the bottler should: A) reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea. B) reject the null hypothesis and conclude that bottles do not contain an average of 16 ounces of tea. C) not reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea. D) not reject the null hypothesis and conclude that bottles do not contain an average of 16 ounces of tea. Your answer: A was incorrect. The correct answer was C) not reject the null hypothesis and conclude that bottles contain an average 16 ounces of tea. Ho: µ = 16; Ha: µ ≠ 16. Do not reject the null since |t| = 1.09 < 1.96 (critical value). Also i’d like to ask, should i judge myself solely on how i perform in the “advanced” level tests of Schweser pro Q bank or is that being too hard on oneself (cuz i cant seem to score more than 60 on those) Appreciated, thanks.

You assume the bottles have 16 ounces (H0), and you try to disprove that (Ha).

But isn’t the Null usually what you DO NOT WANT to prove ? Why would he NOT want the bottles to have 16 ounces ? Is there any properly worded rule for framing the Null hypothesis that you could quote for the above ? Thanks.

Yes, DO NOT WANT to prove = disprove. So, you assume there are 16 ounces in each bottle, unless shown otherwise. They only way to cast doubt on that is by negating it, or showing it is false. Your job is not to prove that it has 16 ounces. That’s not your task in hypotheis testing.

I share Darius’ point I think that H1 = Alternative hypothesis (what we want to prove) H0 = Null hypothsis (what we want to reject) am I wrong ?

kay then its like “to make sure you have what you WANT(16 oz) you should try to DISPROVE it(NO 16 oz)” and the null should then be what you WANT to prove ? I get Dreary’s point… but im not convinced to be honest. so its pointless to argue beyond this perhaps. Thanks for the help folks, im’a let this one pass.

That’s correct, but: > Alternative hypothesis (what we want to prove) You are not trying to prove anything with hypothesis testing. You either reject or not reject a claim.

Hypothesis: “AF Forum is a forum full of useful CFA discussions”. This is a claim that someone is making. What can you do about that? You surely can’t prove it because to do that you have to go and review every post that has ever been made on this forum and make sure it contains a useful discussion, no way to do that. But you can reject this claim by simply showing one post (one post is enough) which does not contain a useful CFA discussion. Thus: H0: AF Forum is full of useful CFA discussions. Ha: AF Forum is not full of useful CFA discussions (i.e., it is not 100% full of …) Since you came up with the bad post, you conclude that you reject the null (H0) in favor of Ha. What if you couldn’t find a bad post? You would then conclude that you cannot reject H0, i.e., you could not justifiably poke any holes in that claim. Homework: 1) What’s a type I error in this example? Think of how it could apply here. 2) What’s a type II error?

The HA is what we want to prove and in this situation we seem to want to keep the mean near 16, not prove is different from 16. Hmm… Which is which here? The answer is that this kind of hypothesis testing is the wrong tool for this job and you need to use some statistical process control method.

Dreary - You are testing a different kind of animal in your example than above (and your hypothesis is really P(post is useful) = p = 1 vs p < 1). In the case above if we use hypothesis testing in some std waay, we come up with a conclusion like “at alpha = 0.05 we can’t reject the hypothesis that mu = 16”. That’s a reaaly messed up way of running a production process - it’s only broken if someone can prove beyond some pretty tough standard that it is broken.

Dreary Wrote: ------------------------------------------------------- > Hypothesis: “AF Forum is a forum full of useful > CFA discussions”. > > This is a claim that someone is making. What can > you do about that? > > You surely can’t prove it because to do that you > have to go and review every post that has ever > been made on this forum and make sure it contains > a useful discussion, no way to do that. > > But you can reject this claim by simply showing > one post (one post is enough) which does not > contain a useful CFA discussion. I think the “Prince Albert” post was one of the best ever (I’m sure JoeyDvivre and some others know what I mean) PS: this post is quite useless too but it gives a feel to what significance level is !

You should not say that you are proving the alternative. You can only prove the alternative by doing an infinite number of tests. So, to prove that the mean is near 16 in the botting example, you could set it up like this: H0: mean is NOT= 16 Ha: mean =16 (this set up is wrong for CFAI purposes) So, you can reject the null easily by finding a sample whose mean =16 +/- some specified std deviation. Doing so, however, does not prove that the mean is 16. It just shows that the claim that the mean is not= 16 is a false claim (we found an instance in which the mean was 16).

Except you can’t do it that way. Remember that you assume H0 is true and then calcuate the probability of observing a sample as contrary to H0 as the one you observed. In this case, there is no sample that is contrary to H0 because not 16 includes numbers like 16.000000000000000001 Edit: Yes I’m still recovering from the Ptince Albert post.

Joey, above it says 16 +/- some specified std deviation, so we are not talking about a continuous distribution issue.

There is something you can do along these lines called “power analysis” but is also beyond the scope - this is just another messed up stats problem. The quality of the statstics instruction in Schweser and CFAI stuff is much much worse than I have seen in any (which is lots and lots) introductory stats book. It’s too bad.

Agreed. The logical concept is clear, but statistical issues can get quite hairy, and I am not good at that.

Interesting discussion. That’s a typicall example of shitty questions. If you don’t reject H0 (thats here the case), you cannot conclude that it’s right. And you cannot conclude that it’s wrong. Why? You dont know the probabilty of Type II error, which could have ex ante any value between 0 and 100 %. So this question shows that the prepprovider is quite often providing silly answers. The Right answer would be: non of A-D.