For the distribution with unknown θ

\(f(x,\theta ) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{\theta };0 \le x \le \theta }\\ {0;elsewhere} \end{array}} \right.\)

We set the testing of hypothesis H₀ ∶ θ = 1 vs H₁ ∶ θ = 2. When the critical region X ≥ 0.4, the value of probability of type-II error is:

Question

Question

This question was previously asked in

SSC CGL Tier-II ( JSO ) 2021 Official Paper ( Held On: 10 August 2022 )

Answer 1

The probability of type-II error is 0.20.

Key Points

In a hypothesis test, the probability of type-II error, also known as β
It is the probability of failing to reject the null hypothesis when it is false.
In other words, it is the probability of making a false negative.
The type-II error rate depends on the choice of the critical region and the value of the alternative hypothesis.

Additional Information

The distribution function has an unknown parameter, θ.
The hypothesis test compares two values of \(θ: H0: θ = 1 and H1: θ = 2.\)
The critical region is defined as \(X ≥ 0.4.\)
In hypothesis testing, a trade-off exists between the probabilities of type-I and type-II errors.
As one decreases, the other increases.
The critical region can be chosen to minimize the probability of type-II error for a given level of type-I error or vice versa.
The choice of critical region and hypothesis test should be made based on the specific problem and desired outcomes.