Question

Let \( X_1 \) and \( X_2 \) be a random sample of size 2 from a discrete distribution with the probability mass function \[ f(x; \theta) = P(X = x) = \begin{cases} \theta, & x = 0, \\ 1 - \theta, & x = 1, \end{cases} \] where \( \theta \in \{0.2, 0.4\} \) is the unknown parameter. For testing \( H_0: \theta = 0.2 \,\,\text{against}\,\, H_1: \theta = 0.4, \) consider a test with the critical region \[ C = \{(x_1, x_2) \in \{0, 1\} \times \{0, 1\} : x_1 + x_2 < 2\}. \] Let \( \alpha \) and \( \beta \) denote the probability of Type I error and power of the test, respectively. Then \( (\alpha, \beta) \) is 
 

• A. \( (0.36, 0.74) \)
• B. \( (0.64, 0.36) \)
• C. \( (0.05, 0.64) \)
• D. \( (0.36, 0.64) \)

Hint:

To find the probability of Type I and Type II errors, use the critical region and compute the probabilities of the test statistics under both \( H_0 \) and \( H_1 \).

Answer 1

The Correct Option is D

Step 1: Find Type I Error $\alpha = P(\text{Reject } H_0 | H_0 \text{ true})$

Under $H_0$: $\theta = 0.2$, so $P(X = 0) = 0.2$ and $P(X = 1) = 0.8$.

$$\alpha = P((X_1, X_2) \in C | \theta = 0.2)$$

$$= P(X_1 = 0, X_2 = 0) + P(X_1 = 0, X_2 = 1) + P(X_1 = 1, X_2 = 0)$$

$$= (0.2)(0.2) + (0.2)(0.8) + (0.8)(0.2)$$

$$= 0.04 + 0.16 + 0.16 = 0.36$$

Step 2: Find Power $\beta = P(\text{Reject } H_0 | H_1 \text{ true})$

Under $H_1$: $\theta = 0.4$, so $P(X = 0) = 0.4$ and $P(X = 1) = 0.6$.

$$\beta = P((X_1, X_2) \in C | \theta = 0.4)$$

$$= P(X_1 = 0, X_2 = 0) + P(X_1 = 0, X_2 = 1) + P(X_1 = 1, X_2 = 0)$$

$$= (0.4)(0.4) + (0.4)(0.6) + (0.6)(0.4)$$

$$= 0.16 + 0.24 + 0.24 = 0.64$$

Answer: (D) $(0.36, 0.64)$