Step 1: Concept Overview:
The size of a critical region (or test), denoted as \(\alpha\), represents the probability of incorrectly rejecting the null hypothesis \(H_0\) when it is true (Type I error). In this scenario, the test relies on the maximum order statistic, \(X_{(4)}\).
Step 2: Core Formulas:
1. The test size is defined as: \( \alpha = P(\text{Reject } H_0 | H_0 \text{ is true}) \).2. Specifically, \( \alpha = P(W_0 | \theta=1) = P(X_{(4)}<1/2 \text{ or } X_{(4)}>1 | \theta=1) \).3. To proceed, we require the distribution of the maximum order statistic, \(X_{(n)}\), derived from a U(0,\(\theta\)) sample. Its CDF is given by \( F_{X_{(n)}}(y) = [F_X(y)]^n \), where \(F_X(y) = y/\theta\) is the CDF of a single U(0,\(\theta\)) variable.
Step 3: Detailed Explanation:
Let's examine the critical region \(W_0\) under the null hypothesis \(H_0: \theta=1\).If \(\theta=1\), all observations \(x_i\) fall within the interval (0, 1). Therefore, the maximum observation, \(x_{(4)}\), must also be less than 1.This implies that the event \(x_{(4)}>1\) is impossible when \(H_0\) is true.Thus, \( P(X_{(4)}>1 | \theta=1) = 0 \).The test size then simplifies to:\[ \alpha = P(X_{(4)}<1/2 | \theta=1) + P(X_{(4)}>1 | \theta=1) = P(X_{(4)}<1/2 | \theta=1) + 0 \]Now, let's determine the distribution of \(X_{(4)}\) under \(H_0\).For a single observation \(X_i\) from U(0,1), the CDF is \( F_X(y) = y \) for \(0 \le y \le 1\).For a sample of size \(n=4\), the CDF of the maximum order statistic \(X_{(4)}\) is:\[ F_{X_{(4)}}(y) = [F_X(y)]^4 = y^4, \text{ for } 0 \le y \le 1 \]The size \(\alpha\) is the probability that \(X_{(4)}\) lies within the critical region, requiring us to compute \( P(X_{(4)}<1/2) \). This is obtained directly from the CDF of \(X_{(4)}\) evaluated at \(y=1/2\).\[ \alpha = F_{X_{(4)}}(1/2) = (1/2)^4 \]\[ \alpha = \frac{1}{16} \]
Step 4: Solution:
The size \(\alpha\) of the critical region \(W_0\) is \( \frac{1}{16} \).