Question about null hypothesis

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Shaz12

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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.
Can anyone tell me what the null hypothesis is?In my opinion it should be u≠16 but the answer states u=16.Isnt the null hypothesis the opposite of what we really want to prove?
 
Presumably, the bottler wants to ensure that at least 16 ounces of tea are in each bottle.
In that case, they want to disprove that there is less than 16 ounces of tea in each bottle, so the null hypothesis would be h0: μ ≤ 16 oz. In that case, the alternative hypothesis is ha: μ > 16 oz.
 
I’m currently studying the same topic, for me the Null Hypothesis depends on what you want to prove.
In your exemple your Null Hypothesis would have been good if the objectif was to prove the average of each bottle was different than 16.
I might be wrong because it’s a tough subject so if someone can confirm or infirm it would be nice
 
It really doesn’t matter which is which.
The Null could be mu <= 16 with Alt equal to mu > 16 or equivalently, the Null could be >= 16 and Alt mu < 16.
Let’s assume the first version is used the the null is rejected.
Then if we examine the problem using hte second version, we would fail to reject the null, reaching an equivalent conclusion.
Some statistics texts claim that the Alt should state what you are trying to prove but this is a but dated (imho) and not really necessary.
 
On a related note, you might be able to determine which null the question is assuming by examining the answer choices.
 
no guys, based on the information presented above only, the researcher wants to ensure an average 16 ounces per bottle, so based on this, it is a two-tailed test, and the null hypothesis (H0) is u≠16
So Shaz 12 is correct, unless you ommitted vital information from the readings.
 
Oh no! I didn’t even read the initial post - I was commenting on @djego’s comment but @olajidean is totally right! Apologies.
 
Shaz12 wrote:
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.
Can anyone tell me what the null hypothesis is?In my opinion it should be u≠16 but the answer states u=16.Isnt the null hypothesis the opposite of what we really want to prove?
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You could argue that the company wants to prove they need to tune up the filling process (before they waste resources on a tune up), meaning that they aren’t filling 16 ounces on average. If this weren’t accurate, why would they do a hypothesis test at all?
Think of the null hypothesis as the assumed state of nature, a starting point for the distribution’s center as defined by the mean (in this example). In this case, the manufacturer makes bottles that claim to hold 16 ounces. On average, they can’t cheat the customer out of product (less than 16, on average), nor do they want to give away “free” product by averaging more than 16 ounces per bottle.
Their assumption is that they are filling 16 ounces, on average, no more, no less (really, they are assuming that the distribution of bottle filling amounts is centered around a mean of 16 ounces, Ho). However, they do some sampling to see if data disagree with this assumption–that is, they decide to look for evidence that might support the idea that, on average, they are bottling something different than 16 ounces (i.e. the distribution’s mean differs from 16 ounces,Ha).
Further, the null doesn’t make sense if we say Ho: mu not equal to 16— can you tell me what this implies about the assumed mean for the distribution of fill amounts? Once you think about that, it should also be more clear that sampling won’t allow us to judge if the data are in a disagreement with our null hypothesis (and why we are setting the null equal to something regardless of the question’s wording).
Hope this helps!
 
S2000magician wrote:
Presumably, the bottler wants to ensure that at least 16 ounces of tea are in each bottle.
In that case, they want to disprove that there is less than 16 ounces of tea in each bottle, so the null hypothesis would be h0: μ ≤ 16 oz. In that case, the alternative hypothesis is ha: μ > 16 oz.
Click to expand...
They don’t want to give it away, either! Ha: mu not equal 16 ounces
 
bujay wrote:
It really doesn’t matter which is which.
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It sort of does matter, from a logical perspective. Ho is the state that is assumed to be true. A lack of evidence to support Ha doesn’t indicate proof of the null.
bujay wrote: The Null could be mu <= 16 with Alt equal to mu > 16 or equivalently, the Null could be >= 16 and Alt mu < 16.
Click to expand...
Agreed that the null could be written either way, but this question stem doesn’t allow for that as written. Additionally, the two null hypotheses are not equivalent.
bujay wrote: Let’s assume the first version is used the the null is rejected.
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Okay. So, we have sufficient evidence to indicate the true mean exceeds 16.
bujay wrote: Then if we examine the problem using hte second version, we would fail to reject the null, reaching an equivalent conclusion.
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But, the conclusion isn’t equivalent, for two reasons. First, FTR Ho does not constitute evidence for Ho. Second, even if it did constitute evidence for Ho (again, it doesn’t), the null is that mu is at least 16, which isn’t equivalent to mu exceeding 16 (especially in this problem). The bottom line, though, is that failure to reject the null does not constitute evidence of the null. It only shows a lack of evidence to support the alternative hypothesis (lack of evidence to indicate that mu is less than 16). The two scenarios you provided are not equivalent.
bujay wrote: Some statistics texts claim that the Alt should state what you are trying to prove but this is a but dated (imho) and not really necessary.
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In your opinion, what makes it dated?
 
Let me start over - next time I will be sure to read the thread more carefully. Firstly, the “equality” always goes in the null hypothesis. This is a rule - not sure if the rationale - but I am sure if the rule. So, valid null hypotheses take the form of =, <=, or >=. You would never have a null of “not equal”. Further, since the “equality” portion must be in the null, you can never have an “equality” in the alternative, as a rule, to ensure mutual exclusiveness.
Now, to answer the original question, the null would be Ho: mu = 16 and Ha: mu not equal to 16, this earring to a two tailed test of a mean.
Now, to go back to what I was saying regarding equivalences, consider a hypothesis test based on a sample of data with Ho: mu <= X and Ha: mu > X. If it turns out to be the case that we reject Ho at a chosen level of significance, then we also know, based on the same data sample, that if we performed a similar test but with Ho: mu >= X and Ha: mu < X, we would fail to reject Ho.
No one has ever really convinced me that the alternative should be what you are trying to prove - this is “dated” in the sense that it is “convention” and not necessary. But again, I may be wrong.
 
Tickersu - thanks for your thoughtful responses. I will offer my thoughts on yours as well once I don’t have to type on a phone!
 
tickersu wrote:
S2000magician wrote: Presumably, the bottler wants to ensure that at least 16 ounces of tea are in each bottle.
In that case, they want to disprove that there is less than 16 ounces of tea in each bottle, so the null hypothesis would be h0: μ ≤ 16 oz. In that case, the alternative hypothesis is ha: μ > 16 oz.
Click to expand...
They don’t want to give it away, either! Ha: mu not equal 16 ounces
Click to expand...
I agree.
I was thinking more of governmental regulation: they’d risk being fined if they had too little, but not if they had too much.
 
bujay wrote: Firstly, the “equality” always goes in the null hypothesis. This is a rule - not sure if the rationale - but I am sure if the rule. So, valid null hypotheses take the form of =, <=, or >=. You would never have a null of “not equal”. Further, since the “equality” portion must be in the null, you can never have an “equality” in the alternative, as a rule, to ensure mutual exclusiveness.
Click to expand...
It doesn’t always go in the null, and you could have a null of not equal– it depends on the context. Noninferiority and equivalence testing is an example in medical/pharm research that utilizes these scenarios. For example, the null (assumed to be true before looking at data) might be that the new treatment is inferior to (or not equal to, perhaps) the current standard treatment. As you can see, this has the practical benefit of reducing harm until we have evidence suggesting that the new treatment is at least as good as the current one.
As for the rationale (why usually equals sign in the null), it’s easiest to see in this example, where we want to make an inference about the mean. The null is the assumed truth; Ho:mu=16 is describing the center of the assumed distribution.We will then see how incompatible the data are with this assumed distribution’s center.

bujay wrote:
Now, to go back to what I was saying regarding equivalences, consider a hypothesis test based on a sample of data with Ho: mu <= X and Ha: mu > X. If it turns out to be the case that we reject Ho at a chosen level of significance, then we also know, based on the same data sample, that if we performed a similar test but with Ho: mu >= X and Ha: mu < X, we would fail to reject Ho.
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I understand your example, but it still isn’t correct to say you’re in an equivalent place with this scenario. FTR Ho in the second case is not equivalent to having evidence to support Ha in the first scenario. You cannot make the same inference.
bujay wrote: No one has ever really convinced me that the alternative should be what you are trying to prove - this is “dated” in the sense that it is “convention” and not necessary. But again, I may be wrong.
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Again, what makes it dated? Being an old convention doesn’t make something dated, necessarily, unless it’s begun to lose applicability, generally speaking. I’ll clarify and say that while I’m not heavily for the idea that the alternative is “what you want to prove,” the logic is clear (and might sit better if the expression was “something you want to investigate.”) If you understand that the null is the assumed state of nature (this is definitional, it’s well-accepted), the status quo, the alternative is clearly what you want to prove/investigate. Think– if the null wasn’t defined as the assumed state of the world, why would hypothesis testing make sense as we use it? If we designated the null as what we want to prove there are two issues: first, the null is assumed to be true, so there would be no logical reason to conduct a test. Second, we can’t use lack of evidence for Ha (FTR Ho) as evidence for Ho. There is a reason why inferences are worded to discuss sufficiency of the evidence in favor of Ha instead of Ho.
 
If you practice the CFAI Practice Problems (‘EOC’) for Reading 11 Hypothesis Testing, you will find that the correct answer states the null hypothesis as the one you want to reject and the alternative hypothesis as the one you want to support.
I think that this example applies the convention of having the equal sign in the null.
 
Moonborne wrote:
If you practice the CFAI Practice Problems (‘EOC’) for Reading 11 Hypothesis Testing, you will find that the correct answer states the null hypothesis as the one you want to reject and the alternative hypothesis as the one you want to support.
I think that this example applies the convention of having the equal sign in the null.
Click to expand...
This isn’t necessarily true. It’s more used as a rule-of-thumb (usually can get you the correct answer).
 
This is precisely why I mentioned the “convention” of using what you are trying to prove in the alternative. That being said, tickersu makes some really interesting points - and clearly knows much more than I do on this topic.
 
I would also add that even though tickersu seems to be correct, I really think you can rely on the “quasi” rule of always having the equality in the null for exam purposes, even though I suspect that tickersu is correct.
 
bujay wrote:
I would also add that even though tickersu seems to be correct, I really think you can rely on the “quasi” rule of always having the equality in the null for exam purposes, even though I suspect that tickersu is correct.
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As far as I’m concerned, you’re absolutely correct that the rule is good for the CFAI exams.
I just made the point because the real world goes far beyond the CFAI curriculum’s breadth (and therefore, “always” becomes untrue, and a further understanding is helpful). This might also be helpful for someone who analyzes healthcare in the case that they are evaluating research for a new drug, for example (the kinds of testing I mentioned above).
 
S2000magician wrote:
tickersu wrote:
S2000magician wrote: Presumably, the bottler wants to ensure that at least 16 ounces of tea are in each bottle.
In that case, they want to disprove that there is less than 16 ounces of tea in each bottle, so the null hypothesis would be h0: μ ≤ 16 oz. In that case, the alternative hypothesis is ha: μ > 16 oz.
Click to expand...
They don’t want to give it away, either! Ha: mu not equal 16 ounces
Click to expand...
I agree.
I was thinking more of governmental regulation: they’d risk being fined if they had too little, but not if they had too much.
Click to expand...
In this day and age, someone might sue because the manufacturer had more calories and sugar in the average bottle than the label claims …
 
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