TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW07.4

Type I error and Type II error

Let $X_{1},\dots ,X_{16}$ be i.i.d. random variables with $X_{1}\sim {\mathcal {N}}(0,4)$. Assume that for a realization it holds ${\bar {x}}=4$. In the context of a right-sided $z$-test, let $H_{0}:\mu =2$ and the rejection area $R=[3,+\infty )$. Which of the following statements are correct?

(a) We will commit a Type I error

(b) We will commit a Type II error

(c) We will not commit a Type II error.

(d) If we increase the significance level of the test, then we obtain a higher test power

Hint: The expectation of $X_{1}$ is fixed at zero. Is the null hypothesis true?

Remark: Note that this is a theoretical consideration of the $z$-test. Here, the distribution of $X_{1}$ is fixed (as the expectation is zero), while usually we assume the expectation (resp. the population) to be unknown. In this theoretical consideration we know whether the null hypothesis is true and we are thus able to make statements about the errors. From a practical point of view, the setup is not useful, because the reason to perform a statistical test is the fact that the population of interest is unknown.

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Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

a) We will commit a Type I error

A Type I error occurs, when the null hypothesis holds true but we reject it. Since we know the underlying distribution we can confidently say that the null hypothesis does not hold true, therefore we also cannot commit a Type I error.

b) We will commit a Type II error

A Type II error occurs, when the null hypothesis does not hold true but we accept it anyway. We know that the null hypothesis does not hold true, and we correctly reject it, so we cannot make a Type II error

c) We will not commit a Type II error.

This is true. We know that the null hypothesis does not hold true, and we correctly reject it

d) If we increase the significance level of the test, then we obtain a higher test power.

The test power is the probability that the null hypothesis does not hold true and we correctly reject it. If we increase significance, the rejection area would increase and therefore the cutoff value for rejection would decrease, which would also increase the test power.