Question

If, \(x \ge 1\) is the critical region for testing \(H_0: \theta = 2\) against the alternate \(H_1: \theta = 1\). On the basis of a single observation from the population \(f(x;\theta) = \theta e^{-x\theta}; x>0, \theta>0\), then the size of Type II error is:

• A. \( \frac{1}{e} \)
• B. \( \frac{1}{e^2} \)
• C. \( \frac{e-1}{e} \)
• D. \( 1 - \frac{1}{e^2} \)

Hint:

Remember the relationship between the critical region and the acceptance region. The acceptance region is always the complement of the critical region. \(\beta\) is the probability of the outcome falling in the acceptance region, calculated under \(H_1\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A Type II error occurs when we fail to reject the null hypothesis (\(H_0\)) when the alternative hypothesis (\(H_1\)) is actually true. The probability of a Type II error is denoted by \(\beta\).

Step 2: Key Formula or Approach:
\(\beta = P(\text{Fail to Reject } H_0 | H_1 \text{ is true})\). The critical (rejection) region is given as \(x \ge 1\). Therefore, the acceptance (fail to reject) region is its complement, which is \(x<1\). We need to calculate \(P(X<1)\) under the assumption that \(H_1\) is true, i.e., \(\theta = 1\).

Step 3: Detailed Explanation:
The population distribution is an exponential distribution with rate parameter \(\theta\). Under the alternative hypothesis, \(H_1: \theta = 1\). The probability density function (PDF) is: \[ f(x; 1) = 1 . e^{-x . 1} = e^{-x}, \quad x>0 \] The probability of a Type II error, \(\beta\), is the probability of the observation falling into the acceptance region \(x<1\), given that \(\theta=1\). \[ \beta = P(X<1 | \theta=1) \] We calculate this by integrating the PDF under \(H_1\) over the acceptance region: \[ \beta = \int_{0}^{1} e^{-x} \,dx \] \[ = [-e^{-x}]_{0}^{1} \] \[ = (-e^{-1}) - (-e^{-0}) = -e^{-1} - (-1) = 1 - e^{-1} \] This can be written as: \[ \beta = 1 - \frac{1}{e} = \frac{e-1}{e} \]
Step 4: Final Answer:
The size of Type II error is \( \frac{e-1}{e} \).