Step 1: Concept Overview:
The power of a statistical test represents the likelihood of correctly rejecting the null hypothesis (\(H_0\)) when the alternative hypothesis (\(H_1\)) is actually true. This is determined by calculating the probability of an observation falling within the critical region, based on the parameter value specified by the alternative hypothesis.
Step 2: Core Formula:
Power is defined as \(P(\text{Reject } H_0 | H_1 \text{ is true})\). Given a critical region of \(1 \le x \le 1.5\), the test's power is \(P(1 \le X \le 1.5)\), calculated using the distribution under \(H_1\).
Step 3: Detailed Calculation:
Under the alternative hypothesis \(H_1: \theta = 2\), the population's probability density function (PDF) is:
\[ f(x; 2) = \begin{cases} \frac{1}{2} & ; 0 \le x \le 2
0 & ; \text{otherwise} \end{cases} \]
This represents a uniform distribution over the interval [0, 2].
The critical region for rejecting \(H_0\) is \(1 \le x \le 1.5\). The test's power is the probability of observing a value \(x\) within this region, given that \(\theta=2\).
\[ \text{Power} = P(1 \le X \le 1.5 | \theta=2) \]
This probability is found by integrating the PDF under \(H_1\) across the critical region:
\[ \text{Power} = \int_{1}^{1.5} f(x; 2) \,dx = \int_{1}^{1.5} \frac{1}{2} \,dx \]
\[ = \frac{1}{2} [x]_{1}^{1.5} \]
\[ = \frac{1}{2} (1.5 - 1) = \frac{1}{2} (0.5) = \frac{1}{4} \]
Step 4: Solution:
The power of the test is \( \frac{1}{4} \).