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A hypothetical statistical hypothesis test that can be used for any type of hypothesis is conducted by drawing a random number between 0 and 1 and rejecting the null hypothesis if it is less than 0.05, what are the type I and type II errors of this test?

Type I error = Pr(reject H0 | H0 true). Now, Pr(reject H0) = 0.05, is this the type I error? How can I calculate type I error without specifying an actual H0?

Type II error = Pr(not reject H0 | H0 false) = 1 - Pr(reject H0 | H0 false). Does this equal to Pr(not reject H0) = 0.95?

Please help.

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    $\begingroup$ "How can I calculate type I error without specifying an actual H0?" You have an explicit rejection rule and (since it doesn't consider the hypothesis at all) you can easily calculate the probability of rejection happening when $H_0$ it true. $\endgroup$
    – Glen_b
    Commented Sep 30, 2022 at 4:58

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You don't say it explicitly, but assume that the random number between 0 and 1 has a uniform distribution.
Let's be more realistic and assume that you hope that it's uniform, but not sure, so you take it as a null hypothesis and then follow you path. But if H0 is false and the distribution isn't uniform, you can say absolutely nothing about probabilities. That's the problem common to all "frequentist" statistics.

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  • $\begingroup$ yes, the random number generation has a Uniform[0,1] distribution,, given this would we be able to calculate the type i and ii error rates? $\endgroup$
    – Cabbage Roll
    Commented Sep 30, 2022 at 0:32
  • $\begingroup$ You either hypothesize that the distribution is uniform (and this is your H0) or state that it can't be non-uniform and select another H0. What is your H0? There can be no error (of any type) if there is no hypothesis. $\endgroup$
    – Alex
    Commented Sep 30, 2022 at 0:37
  • $\begingroup$ Got it, thanks! $\endgroup$
    – Cabbage Roll
    Commented Sep 30, 2022 at 4:49
  • $\begingroup$ "Can say absolutely nothing" is too exaggerated to be worth considering seriously. Classical statistical theory and practice were never that limited. $\endgroup$
    – whuber ♦
    Commented Sep 30, 2022 at 13:06

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