Question

If, \(1 \le x \le 1.5\) is the critical region for testing the null hypothesis \(H_0: \theta=1\) against the alternative hypothesis \(H_1: \theta=2\) on the basis of a single observation from the population, \( f(x;\theta) = \begin{cases} \frac{1}{\theta} & ; 0 \le x \le \theta \\ 0 & ; \text{otherwise} \end{cases} \), then the power of the test, is

• A. \( \frac{3}{4} \)
• B. \( \frac{1}{2} \)
• C. \( \frac{4}{5} \)
• D. \( \frac{1}{4} \)

Hint:

To calculate the power of a test, always use the parameter value from the alternative hypothesis (\(H_1\)). To calculate the size of the test (Type I error, \(\alpha\)), use the parameter value from the null hypothesis (\(H_0\)).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The power of a statistical test is the probability of correctly rejecting the null hypothesis (\(H_0\)) when the alternative hypothesis (\(H_1\)) is true. It is calculated as the probability of the observation falling into the critical region, assuming the parameter value from the alternative hypothesis.

Step 2: Key Formula or Approach:
Power = \(P(\text{Reject } H_0 | H_1 \text{ is true})\) Given the critical region is \(1 \le x \le 1.5\), the power of the test is \(P(1 \le X \le 1.5)\) calculated using the distribution under \(H_1\).

Step 3: Detailed Explanation:
Under the alternative hypothesis, \(H_1: \theta = 2\). The probability density function (PDF) of the population is: \[ f(x; 2) = \begin{cases} \frac{1}{2} & ; 0 \le x \le 2
0 & ; \text{otherwise} \end{cases} \] This is a uniform distribution on the interval [0, 2]. The critical region for rejecting \(H_0\) is given as \(1 \le x \le 1.5\). The power of the test is the probability that the observation \(x\) falls within this region, given that \(\theta=2\). \[ \text{Power} = P(1 \le X \le 1.5 | \theta=2) \] We calculate this probability by integrating the PDF under \(H_1\) over 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: Final Answer:
The power of the test is \( \frac{1}{4} \).