Let X1, X2, …, X10 denote a random sample of size 10 from a...

Let $X_{1}, X_{2}, \ldots, X_{10}$ denote a random sample of size 10 from a Poisson distribution with mean $\theta .$ Show that the critical region $C$ defined by $\sum_{1}^{10} x_{i} \geq 3$ is a best critical region for testing $H_{0}: \theta=0.1$ against $H_{1}: \theta=0.5 .$ Determine, for this test, the significance level $\alpha$ and the power at $\theta=0.5$.

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the count of independent events occurring in a fixed interval of time or space given a constant mean rate. It is particularly useful in scenarios where the events happen at a relatively low rate and the occurrences are independent from one time interval to the next.

Neyman-Pearson Lemma

The Neyman-Pearson lemma is a foundational result in hypothesis testing that provides a systematic way to construct the most powerful test for distinguishing between two simple hypotheses. It justifies the use of the likelihood ratio test as the optimal test in terms of maximizing power for a given significance level.

Likelihood Ratio Test

The likelihood ratio test compares the likelihood of the observed data under the alternative hypothesis to that under the null hypothesis. By setting a critical value, the test determines whether to reject the null hypothesis, making it a central tool in constructing tests with optimal properties as per the Neyman-Pearson lemma.

Critical Region

The critical region is the set of values of the test statistic that leads to the rejection of the null hypothesis. In hypothesis testing, selecting a critical region is crucial as it determines the conditions under which the null hypothesis is discarded, thereby controlling the probability of a Type I error.

Significance Level

The significance level, denoted by alpha, is the probability of incorrectly rejecting the null hypothesis when it is in fact true (a Type I error). It sets the threshold for how extreme the data must be before the null hypothesis is rejected.

Power of a Test

The power of a test is the probability that it correctly rejects the null hypothesis when the alternative hypothesis is true. It reflects the test's ability to detect an effect or difference when one actually exists, making it a critical measure of a test’s effectiveness.

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