Suppose $X_n$ is an i.i.d. random sample from the $N(μ,σ^2)$ population, where μ is unknown but $σ^2$ is known. Consider a test statistic $T =\sqrt{n}('bar{X}_n − μ_0)/σ$ at the significance level α for $H_0 :μ=μ_0\quad versus\quad H_a :μ\neqμ_0$. Find Type I error of this test.


One of my friends gives this answer:

The hypothesis model will be



Type I error $\alpha$ is the probablity of rejecting the null hypothesis $H_o$.

Thus $p=\alpha$

I think it's not correct, because Type I error is the the probablity of rejecting the null hypothesis $H_o$ when $H_o$ is true , and $\alpha$ means the size of test is no larger than $\alpha$, so I think there are some differences between these two concepts.

So who is wrong? why? and what the correct answer should be?


If you say that your test has significance level of $\alpha$, it is by definition means that the probability of rejecting the null hypothesis under $H_0$ is $\alpha$. Namely, you said that your test is of the following form $$ \varphi(X)=I\{|T|\ge t^{(n-1)}_{\alpha/2}\}, $$ thus $$ \mathbb{P}\left(\varphi(X)=1| H_0:\mu =\mu_0 \right)=\alpha. $$ The "size" of a test is the probability to reject the null hypothesis under $H_0$. When you have built your test statistic you have actually started from the desired size and then derived the statistic. The other way around is meaningful only when you have some rejection rule, i.e., reject $H_0$ when such and such happens. Then you can find its size, otherwise you start with the desired "size" and proceed to the construction of the rule (the rejection region).


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