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While studying about the hypothesis testing, I encountered the following case -

Consider a case where we are carrying out hypothesis testing for amount of perfume filled in the bottles:

NULL hypothesis = bottle filling machine fills bottles with 150 cc of perfume in each bottle

Alternative hypothesis = bottle filling machine is NOT filling 150 cc of perfume in each bottle

And we know that the amount of perfume filled in the bottle by the machine has a normal distribution with mean=150 and standard deviation = 2.

Now my question is -

If we know that amount has distribution with standard deviation = 2, then what is the need to carry out the hypothesis test?

Why can't we directly reject the NULL hypothesis? Since everything has a variance, and in this case, the amount of perfume filled in the bottle will too have a variance.

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    $\begingroup$ Usually the null hypothesis is something like "bottle filling machine fills bottles with an average of 150 cc of perfume in each bottle" or "bottle filling machine fills at least 95% of bottles with at least 150 cc of perfume in each bottle". $\endgroup$
    – Henry
    Aug 24, 2023 at 10:33
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    $\begingroup$ The point of hypothesis testing is to deal with the case that you do not "know" the precise distribution, or you want to check whether it has changed $\endgroup$
    – Henry
    Aug 24, 2023 at 10:36

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The claim seems to be that the machine fills with $150$ every time. Thus, any measurement that is not $150$ disproves the claim. Since there is positive variance to the amount of filling, the amount of filling varies, meaning that, at least once, the machine fills with a quantity other than $150$.

This is absolute proof that the claim is incorrect, essentially finding a counterexample like we do in pure mathematics, rather than hypothesis testing in statistics.

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The problem is poorly worded (not by you, but by whoever wrote the "case" that you discuss).

Two people (in comments and answers) have identified some of the problems with the problem: 1) That's not how hypothesis tests are usually worded 2) You rarely know what the machine does, exactly and 3) The claim, as stated, is exact, and easily refuted by an example, with no statistics needed.

We sometimes do know the population parameters. That is when we have data for the entire population. But that can't apply here, as we will, presumably, want to keep using the machine in the future.

Also, the null hypothesis, as stated, contradicts what we "know" about the machine. That is, that the amount of perfume in bottles varies.

Do you know who wrote this problem?

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