Question

Let \(X_{1}, X_{2}\) be a random sample of size \(n=2\) from the distribution having pdf \(f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0

Short answer

The significance level (alpha) of the test is 0.8 and the power of the test (1-beta) is 0.94.

Step-by-step solution

- Derive the test's decision rule (ratio)

In the null hypothesis the pdf is \(f(x;2) = 1/ 2 e^{-x / 2}\) and in the alternate hypothesis, it is \(f(x;1) = e^{-x}\). Substitute the values in the the given condition, to derive a new condition. \(\frac{f(x_{1};2)f(x_{2};2)}{f(x_{1};1)f(x_{2};1)} \leq 1 / 2 \rightarrow \frac{1 / 4 e^{-x_{1} / 2} e^{-x_{2} / 2}}{e^{-x_{1}} e^{-x_{2}}} \leq 1 / 2 \rightarrow 4 e^{x_{1} / 2 + x_{2} / 2} \leq e^{x_{1} + x_{2}} \rightarrow e^{x_{1} / 2 + x_{2} / 2} \leq 1 / 4 \rightarrow x_{1} + x_{2} \geq 2 * ln(4) \rightarrow x_{1} + x_{2} \geq 2.77\)

- Find the critical region

The critical region, C, for the rejection of the null hypothesis is the set of values that meet the condition. So, the critical region is \(C = \{ (x_1, x_2): x_1 + x_2 \geq 2.77 \}\) where x_i is the observed value of \(X_i\).

- Calculate the significance level of the test

The significance level, alpha, is the probability that the null hypothesis is true but gets rejected, which uses integral calculus as \(\alpha = P_{\theta=2} (X_1 + X_2 \geq 2.77)\). Using the exponential distribution property, this is equivalent to \(1 - P_{\theta=2} (X_1 + X_2 < 2.77)\), which results to \(\alpha = 1 - e^{-2.77 / 2}= 0.8\)

- Calculate the power of the test

The test's power (1-beta) is the probability that the alternate hypothesis is true, and it is accepted. It is calculated as \(1- \beta = P_{\theta=1} (X_1 + X_2 \geq 2.77)\). Similar to the above step, this becomes \(1 - P_{\theta=1} (X_1 + X_2 < 2.77)\), which yields \(1 - \beta = 1 - e^{-2.77} = 0.94\)