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I recently saw this question in some Statistics problems I was doing:

A jug of Fun Juice is supposed to contain at least 64 oz of juice. The jug-filling machine is calibrated so that the average bottle should contain 64.05 oz of juice. A manager wants to know if the machine is under-filling the bottles. She plans to take a random sample of bottles and then perform a significance test.

What are the null and alternative hypotheses?

The answer key states that $H_0$ is $\mu=64.05$ and $H_a$ is $\mu<64.05$. However, I got that $H_0$ is $\mu=64$ and $H_a$ is $\mu<64%$. Wouldn't "underfilling" here mean that the mean is below what it is supposed to contain (64 ounces) and not below what the machine is calibrated to fill (64.05 ounces)? Is this an error in the textbook, or am I just missing something?

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  • $\begingroup$ You may find this answer helpful. $\endgroup$
    – dimitriy
    Commented Jun 13 at 3:44
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    $\begingroup$ The crux of the matter is that "underfilling" has two reasonable but different interpretations: it might mean "not achieving the required minimum of 64 oz" or it might mean "averaging less than 64.05." I find the first interpretation to be most relevant from a business perspective, but it lacks precision: what is the acceptable frequency with which bottles might have less than 64 oz of juice? Consequently, as smart test-takers we ought to guess the second interpretation is intended because it leads to the simple, thoughtless textbook solution; but your thinking appears to be more appropriate. $\endgroup$
    – whuber ♦
    Commented Jun 13 at 13:43

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By calibrating the machine so that the average fill is 64.05, the manager is effectively saying that with this setting the proportion of underfilled bottles would be just acceptable. So manager really wants to know if the machine is now under-filling too many bottles, or is it still calibrated correctly.

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  • $\begingroup$ This doesn't seem to be an answer. What would be the correct answer and the justification? $\endgroup$
    – wyatt400
    Commented Jun 13 at 0:35
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    $\begingroup$ @wyatt400 read the problem as "A jug of Fun Juice is supposed to contain at least 64 oz of juice. In order to get this for all individual bottles, the jug-filling machine is calibrated so that the average bottle should contain 64.05 oz of juice. A manager wants to know if the machine is under-filling some of the bottles (ie. the calibration has shifted below the required 64.05 oz)." Testing whether the machine is under-filling and testing whether the machine is correctly set at 64.05 is the same problem. $\endgroup$
    – Sextus Empiricus
    Commented Jun 13 at 12:04
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    $\begingroup$ One can be critical about the question and state that the expression "A manager wants to know if the machine is under-filling the bottles." is not well defined. $\endgroup$
    – Sextus Empiricus
    Commented Jun 13 at 12:06
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Well, the answer key is wrong, in fact horribly wrong. But your thinking process is also wrong; not as wrong as the answer key, but not correct nevertheless.

  1. Why is the answer key wrong? In this case, to prove the machine is doing its job properly, we would need to accept the null proposed by the answer key (over the alternative). But one can never accept the null, just fail to reject it, just like one can never reject the alternative, just fail to accept it. Think of it this way; I will just take 3 samples, the sample size will be so small that the confidence interval of the mean will be quite large, so large that it will include 64.05oz, therefore I will fail to reject the null and declare my machine to be working fine. This is absurd, and very concerning if it really happens to be in a textbook, or even in the handout from your professor. All you can do is disprove the null, never prove it. So the answer key is horribly wrong (because it implies that you will try to "prove" the null).
  2. But your thinking is also incorrect (it is logical, but not practical). If the machine was filling exactly 64oz, on average, that would mean that half the customers would get less than the label says... Not a good situation commercially. In practice, any filling machine always overfills a bit (be it juice, drugs, shampoo, chips, m&m's, etc.), so that a proportion > 50% of containers ends up with no less than the advertised quantity. So we need to show that the mean is >= 64.05oz. Therefore our null must be $H_0: \mu<64.05$ and the alternative $H_a: \mu>=64.05$.
    But, and this is where the problem gets actually interesting, one can not do what I just wrote (even though that is what we really want to do!). I do not know of a single test where the alternative is of the form $>=$, or $<=$. So instead you can pick e.g. $H_0: \mu=64.049$ and the alternative $H_a: \mu>64.049$. Now a simple t-test will give you the answer. But 64.049 may allow the mean to be 64.0495, which is below 64.05?! So we could use 64.0499, or 64.04999, or ... Probably, for all practical purposes, 64.04999 is just as good as 64.05? (our measuring instrument probably does not allow us that much resolution anyway). One could always use 64.05, but our alternative now becomes that the mean is strictly greater than 64.05, and thus the machine is overfilling more than is necessary (even if it is by just a little bit)...
    So the question is doubly wrong; it gives the wrong answer, and I know of no test which can prove that a mean is $>=$ than any given value; only $>$...

Another issue is that the real specification for the mean can not practically be $\mu=64.05$; that is not achievable (except in -not very good- homework assignments), because in the real world, the average fill will vary, day to day, shift to shift, week to week, machine to machine, etc. A more appropriate specification could have been, e.g. $64.05<\mu<64.1$. In that case, the null is $H_0: {\mu<=64.05}$ $or$ ${\mu>=64.1}$ and the alternative $H_a: 64.05<\mu<64.1$. And the solution is to run 2 single sided t-tests (one for each "side" of the null), and sum the p-values.

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  • $\begingroup$ Not just the answer key is wrong, the problem statement is confusing and ambiguous as well. $\endgroup$
    – Sextus Empiricus
    Commented Jun 13 at 12:08
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I understand the confusion but the manager just want to know more about the precision of the machine which is testing the assumption that the calibrated value is equal to dispensed amount in the can is supposed to be exactly 64.05 oz which would make each can contain at least 64 oz.

That's how I would justify the answer of the book in its implied meaning of "underfilling". Then I would reject yours because it was never advertised that the can should contain 64 oz which is what is referred by your null hypothesis.

However, I'm confused more so of the expression of the hypotheses because it looks like a mix formulation between one-tailed tests ($>$ or $<$) and two-tailed tests ($=$ or $\neq$)

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