Let X1, X2 be a random sample of size n=2 from the...

Let $X_{1}, X_{2}$ be a random sample of size $n=2$ from the distribution having pdf $f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0<x<\infty$, zero elsewhere. We reject $H_{0}: \theta=2$ and accept $H_{1}: \theta=1$ if the observed values of $X_{1}, X_{2}$, say $x_{1}, x_{2}$, are such that $$ \frac{f\left(x_{1} ; 2\right) f\left(x_{2} ; 2\right)}{f\left(x_{1} ; 1\right) f\left(x_{2} ; 1\right)} \leq \frac{1}{2} $$ Here $\Omega=\{\theta: \theta=1,2\}$. Find the significance level of the test and the power of the

Let $X_{1}, X_{2}$ be a random sample of size $n=2$ from the distribution having pdf $f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0<x<\infty$, zero elsewhere. We reject $H_{0}: \theta=2$ and accept $H_{1}: \theta=1$ if the observed values of $X_{1}, X_{2}$, say $x_{1}, x_{2}$, are such that
$$
\frac{f\left(x_{1} ; 2\right) f\left(x_{2} ; 2\right)}{f\left(x_{1} ; 1\right) f\left(x_{2} ; 1\right)} \leq \frac{1}{2}
$$
Here $\Omega=\{\theta: \theta=1,2\}$. Find the significance level of the test and the power of the

Key Concepts

Power of the Test

The power of a test is the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true. It measures the sensitivity of the test to detect an effect when one exists, thereby indicating the effectiveness of the test in identifying true positives.

Significance Level

The significance level of a test, also known as the Type I error rate, is the probability of rejecting the null hypothesis when it is actually true. It represents the risk of a false positive and is predetermined by the researcher before the test is conducted.

Likelihood Ratio Test

The likelihood ratio test is a method for comparing two competing statistical models or hypotheses by computing the ratio of their likelihoods given the data. A decision rule based on this ratio determines whether to reject the null hypothesis in favor of the alternative hypothesis, with the threshold often set to control the significance level of the test.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is characterized by a rate or scale parameter, and it uniquely possesses the memoryless property, meaning that future probabilities do not depend on past occurrences.

Hypothesis Testing

Hypothesis testing is a statistical framework used to decide whether there is enough evidence in a sample to support a certain claim (the alternative hypothesis) over another claim (the null hypothesis). This involves calculating test statistics and comparing them to critical values or thresholds at a pre-determined significance level.

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