Question

Let \( X \) be a sample observation from \( U(\theta, \theta^2) \) distribution, where \( \theta \in \Theta = \{2, 3\} \) is the unknown parameter. For testing \[ H_0 : \theta = 2 \,\, \text{against} \,\, H_1 : \theta = 3, \] let \( \alpha \) and \( \beta \) be the size and power, respectively, of the test that rejects \( H_0 \) if and only if \( X \geq 3.5 \). Then \( \alpha + \beta \) equals ............... 
 

Hint:

To find \( \alpha \) and \( \beta \), use the CDF of the distribution at the critical point for both the null and alternative hypotheses.

Answer 1

Correct Answer: 1.1 - 1.2

Step 1: Determine the size \( \alpha \). 
The size of the test is the probability of rejecting \( H_0 \) when \( \theta = 2 \). We reject \( H_0 \) if \( X \geq 3.5 \), so we need to calculate the probability that \( X \geq 3.5 \) under \( H_0 \) (i.e., when \( \theta = 2 \)). For \( X \sim U(2, 4) \), the cumulative distribution function (CDF) is: \[ P(X \geq 3.5 | \theta = 2) = \frac{4 - 3.5}{4 - 2} = \frac{0.5}{2} = 0.25. \] Thus, \( \alpha = 0.25 \).

Step 2: Determine the power \( \beta \). 
The power of the test is the probability of rejecting \( H_0 \) when \( \theta = 3 \). We reject \( H_0 \) if \( X \geq 3.5 \), so we calculate the probability that \( X \geq 3.5 \) under \( H_1 \) (i.e., when \( \theta = 3 \)). For \( X \sim U(3, 9) \), the CDF is: \[ P(X \geq 3.5 | \theta = 3) = \frac{9 - 3.5}{9 - 3} = \frac{5.5}{6} \approx 0.9167. \] Thus, \( \beta = 0.9167 \).

Step 3: Conclusion. 
Therefore, \( \alpha + \beta = 0.25 + 0.9167 = 1.1667 \).