Step 1: Concept Overview:
The power of a test represents the probability of correctly rejecting \(H_0\) when \(H_1\) is true. Given an exponential distribution with mean \(\theta\), we aim to determine the probability of the sample falling within the critical region \(W_0\), assuming the parameter value specified by \(H_1\).
Step 2: Core Formula and Methodology:
1. Power is defined as \(P(\text{Reject } H_0 | H_1 \text{ is true}) = P((X_1, X_2) \in W_0 | \theta = 1)\).
2. This simplifies to \(P(X_1 + X_2 \ge 6.5 | \theta=1)\).
3. The sum of \(n\) independent and identically distributed (i.i.d.) Exponential(\(\lambda\)) random variables follows a Gamma(\(n, \lambda\)) distribution. Since the mean is \(\theta\), the rate parameter is \(\lambda = 1/\theta\).
4. A Gamma variable can be transformed into a Chi-squared variable using the relationship: If \(S \sim \text{Gamma}(k, \lambda)\), then \(2\lambda S \sim \chi^2_{2k}\).
Step 3: Detailed Explanation:
Under the alternative hypothesis \(H_1: \theta = 1\), \(X_1\) and \(X_2\) are i.i.d. exponential random variables with mean \(\theta=1\). The rate parameter is \(\lambda = 1/\theta = 1\).
Let \(S = X_1 + X_2\). Since \(X_1, X_2\) are i.i.d. Exp(1), their sum \(S\) follows a Gamma distribution with shape \(n=2\) and rate \(\lambda=1\). Thus, \(S \sim \text{Gamma}(2, 1)\).
Applying the Chi-squared transformation:
If \(S \sim \text{Gamma}(k=2, \lambda=1)\), then \(2\lambda S = 2(1)S = 2S\) follows a Chi-squared distribution with \(2k = 2(2) = 4\) degrees of freedom.
\[ 2(X_1 + X_2) \sim \chi^2_{(4)} \]
The power of the test is \(P(X_1 + X_2 \ge 6.5)\) under \(H_1\). Transforming the inequality to use the Chi-squared variable:
\[ X_1 + X_2 \ge 6.5 \]
Multiply by 2:
\[ 2(X_1 + X_2) \ge 2(6.5) \]
\[ 2(X_1 + X_2) \ge 13 \]
Since \(2(X_1+X_2)\) is a \(\chi^2_{(4)}\) variable, the power is:
\[ \text{Power} = P(\chi^2_{(4)} \ge 13) \]
Step 4: Final Result:
The power of the test is \( P(\chi^2_{(4)} \ge 13) \).