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A test is made of $H_0: M= 77.4$ and $H_a: M<77.4.$ Assume you reject the null, because you did not feel like doing a hypothesis test, and you are pretty sure your alternative is correct anyway. Have you made a Type 1 error, Type 2 error, a correct decision, or a silly decision?

I keep trying to solve this using the wording but nothing is making sense to me. I’ve already tried to think the the mean is less than $77.4,$ so the $H_0$ is false. But when I got correct decision the server said it was wrong. I also don’t see an option for a silly decision in the given chart.

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  • $\begingroup$ Welcome to Cross Validated! On what chart do you not see “silly decision” as an option? $\endgroup$
    – Dave
    Commented Dec 1, 2022 at 7:42
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    $\begingroup$ For more experienced readers: en.wikipedia.org/wiki/Type_III_error $\endgroup$
    – B.Liu
    Commented Dec 1, 2022 at 8:05
  • $\begingroup$ None of the above, because there is not enough information to evaluate the first three options and the fourth ("silly decision") has no objective meaning. $\endgroup$
    – whuber ♦
    Commented Dec 1, 2022 at 15:47

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In my opinion that's a silly decision because Type 1 error, Type 2 error and a correct decision all assume you know both the "True" hypothesis and the stat that you are measuring. But in this case you are not measuring any stat, therefore I'd go with silly decision, which could be correct or wrong by chance your (random) guess is true or not.

I'd point you to the second comment of B.Liu for more infos

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    $\begingroup$ I think it's a bit extreme and potentially misleading to call the fourth option "silly." We make these kinds of "silly decisions" all the time, with great justification. Plenty of questions don't require any data to resolve. For instance, is the sun greater than 77.4 meters away or not? Indeed, such non-data decisions are routinely involved in the judgment calls made when formulating statistical models: we set reasonable bounds to our sample spaces or we choose prior distributions that imply such bounds, and on and on. $\endgroup$
    – whuber ♦
    Commented Dec 1, 2022 at 17:49
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    $\begingroup$ Well, that's a good point! I think then that the question is ill-posed and if we really have to chose between one of the four options I would totally go with the last one. $\endgroup$
    – DaSim
    Commented Dec 2, 2022 at 14:42
  • $\begingroup$ Just one additional point: in your examples our priors still come from measured data/observed scenarios (e.g. we know the sun is further than 77 m since scientists estimated that measurement + we built higher buildings that did not go into the sun). But I don't think that's the only case. As a weird example, suppose to have an instrument that measures if god exists or not, and to answer only based on your priors (= your faith/education). That answer in my opinion may be considered silly (with silly meaning unsupported by data but just on unproven priors) $\endgroup$
    – DaSim
    Commented Dec 2, 2022 at 14:43
  • $\begingroup$ I'm confident people have known the sun is further than that long before they every conceived of linear measurements: this is a matter of unquantified experience. Relying on that is not "silly." It could be characterized as protophysics. But as a matter of answering a badly written multiple choice question, I can only agree with you that the best strategy would be to select the fourth option. $\endgroup$
    – whuber ♦
    Commented Dec 2, 2022 at 15:21

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