Determine the probability of committing a Type II Error.

statistics

Suppose X is uniformly distributed on the interval $[0;\mu]$, with $\mu$ unknown. The null hypothesis is that $\mu = 2.5$ and the alternative hypothesis is that $\mu \geq 2.5$.
We test the hypothesis by sampling $X_1$ and $X_2$ from $X$ and taking the maximum of the two as our test statistic $T$.
We decide to reject $H_0$ in favor of $H_1$ when $T \geq 2$.
Suppose that the real value of $\mu$ is equal to $3$.
Determine the Probability of committing a Type II Error.

I have some problem to compute the Probability of committing a Type II Error in this exercise, how should I start it, do I have to convert to N(0,1) distribution?

Best Answer

Hint:

Let $\beta$ denote the probability of a type II error under the assumption that $\mu = 3$. This means $T < 2$ although values up to $\mu = 3$ can be assumed. Then

  • $\beta = P_{\mu = 3}(T <2) = P_{\mu = 3}([0,2]\times[0,2])$ where
  • $P_{\mu = 3}$ is the uniform distribution on the square $[0,3]\times[0,3]$.

Some more info:

Note that you are dealing with squares, as the sample consists of two (independent) random variables. So, you need to consider the squares $2^2$ and $3^2$. Then, you get the correct results.

Maybe you may draw the region $T<2$ on the square with side length $3$ to get a visual grip of what you are calculating.

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