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Let $X$ be a pollution index measured close to a nuclear plant. We model it by a $\mathcal{N}\left(\mu, \sigma^{2}\right)$ distribution. The standard deviation is supposed to be known and equal to $4 .$ The state regulations fix the maximal index at $30 .$

The nuclear plant head manager wants to show that his plant complies with the regulations. What hypotheses $H_{0}$ and $H_{1}$ should he test for? Propose an appropriate test. Establish the decision rule for that test at thresholds $5 \%$ and $1 \% .$

I don't know how to choose $H_0$ and $H_1$ in this case, can somebody help please ?

EDIT : I'm not looking for a full fledged solution but just some hints so I can get started.

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    $\begingroup$ Considering the regulations, you can divide the possible values of $\mu$ into two regions, with very different implications when the true value of $\mu$ is in each. One region must correspond to the null hypothesis; one to the alternative: work out how the tests would differ & you can decide which is most appropriate. $\endgroup$ Feb 23, 2020 at 12:04
  • $\begingroup$ The question doesn't quite give enough information. When you say the 'maximal' index is fixed at 30 what does this mean? Clearly if the pollution index is normally distributed then it will sometimes be more than 30, because normal distributions are not bounded. Is it the case that the average should be less than 30, or the index should be less than 30 a certain proportion of the time? Or do you assume the pollution is fixed and the measurement is what has the error? $\endgroup$ Feb 23, 2020 at 14:36
  • $\begingroup$ @GeorgeSavva I really have no idea and I was similarly confused, I found the problem as is, note that I have no problem with the theoretical aspects of hypothesis testing, the very reason I'm asking for help with this problem is because I couldn't translate it into a mathematical one, if I could just know what's $H_0$ and $H_1$, I can definitely handle the rest. $\endgroup$ Feb 23, 2020 at 14:42
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    $\begingroup$ The answer to this question is a legal matter. In US law and US EPA guidance, the choice of null and alternative hypotheses depends on the regulation and the program under which the power plant is monitoring. It's not for statistics to decide! For instance, if this is a groundwater monitoring program under the Resource Conservation and Recovery Act (RCRA) and the plant is monitoring to detect possible new contamination, the null hypothesis is that the concentrations are below the limits; but if the plant is monitoring the progress of a cleanup, the null is that concentrations exceed limits. $\endgroup$
    – whuber
    Feb 23, 2020 at 17:38
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    $\begingroup$ (Continued) AFAIK, all state regulations are modeled after the federal ones and adhere closely to them as far as statistical approaches are concerned. For more about this see 40 CFR 264, 40 CFR 265, and US EPA guidance on statistical methods for RCRA groundwater monitoring (2009 as well as interim guidance from 1989). $\endgroup$
    – whuber
    Feb 23, 2020 at 17:42

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