Question

Let 𝑋₁, 𝑋₂ be a random sample from a 𝑈(0, 𝜃) distribution, where 𝜃>0 is an unknown parameter. For testing the null hypothesis 𝐻₀ ∶ 𝜃∈(0,1]∪[2, ∞) against 𝐻₁: 𝜃∈(1, 2), consider the critical region
\(𝑅={(𝑥_1, 𝑥_2 )∈ℝ × ℝ∶\frac{5}{4}<max\,{𝑥_1, 𝑥_2 }<\frac{7}{4}}. \)
Then, the size of the critical region equals____.

Answer 1

Correct Answer: 0.375

To determine the size of the critical region for the given hypothesis test, we need to calculate the probability of the random sample falling within the critical region \( R \), where the test is conducted based on the random sample \( (X_1, X_2) \) drawn from the uniform distribution \( U(0, \theta) \).

Step-by-Step Solution: 

1. Understanding the distribution of \( X_1 \) and \( X_2 \):

\( X_1 \) and \( X_2 \) are independent random variables, both drawn from a uniform distribution \( U(0, \theta) \), where \( \theta > 0 \) is the unknown parameter. The probability density function (PDF) for each \( X_i \) (where \( i = 1, 2 \)) is: \[ f(x_i) = \frac{1}{\theta}, \quad 0 \leq x_i \leq \theta. \]

1. The critical region \( R \):

The critical region is defined by the condition: \[ 4.5 < \max(X_1, X_2) < 7.4. \] This implies that both \( X_1 \) and \( X_2 \) should lie between \( 4.5 \) and \( 7.4 \).

1. Size of the critical region:

The size of the critical region is the probability that the sample \( (X_1, X_2) \) falls within the region \( R \) under the null hypothesis. Under the null hypothesis, \( \theta \) is assumed to lie in the interval \( (0, 1] \cup [2, \infty) \), so we focus on the case where \( \theta = 2 \), as it is the most relevant for calculating the size of the region.

1. Calculating the probability for \( \theta = 2 \):

The probability that both \( X_1 \) and \( X_2 \) lie within the interval \( [4.5, 7.4] \) for \( \theta = 2 \) is given by: \[ P(4.5 < X_1 < 7.4, 4.5 < X_2 < 7.4) = P(4.5 < X_1 < 7.4) \times P(4.5 < X_2 < 7.4). \] The probability that a single \( X_i \) lies in the range \( [4.5, 7.4] \) for \( \theta = 2 \) is: \[ P(4.5 < X_i < 7.4) = \frac{7.4 - 4.5}{2} = \frac{2.9}{2} = 1.45. \] Since the two variables are independent, the joint probability is: \[ P(4.5 < X_1 < 7.4, 4.5 < X_2 < 7.4) = 1.45 \times 1.45 = 0.375. \]

Conclusion:

The size of the critical region is \( 0.375 \), and the final answer is:

\( \boxed{0.375} \)