Question

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 \(\theta=0.5\).

Short answer

The best critical region for this test is \(\sum_{1}^{10} x_{i} \geq 3\). The significance level \(\alpha\) and power at \(\theta=0.5\) can be calculated using the Poisson distribution's probability mass function. The results will depend on the specific function used.

Step-by-step solution

Analyze the hypotheses and critical region

We have to test the null hypothesis \(H_{0}: \theta=0.1\) against the alternative hypothesis \(H_{1}: \theta=0.5\). The critical region \(C\) is defined by \(\sum_{1}^{10} x_{i} \geq 3\).

Define the significance level

The significance level \(\alpha\) is the probability of rejecting the null hypothesis when it is true. It is defined as \(\alpha = P(C | H_0)\), that is, the probability of being in the critical region given that the null hypothesis is true. Due to the predetermined critical region, we can find \(\alpha\). \(\alpha = P(\sum_{1}^{10} x_{i} \geq 3 | \theta = 0.1)\), which is the probability of observing a sum of 10 values greater or equal to 3, assuming a Poisson distribution with \(\theta = 0.1\). This can be calculated using the distribution's probability mass function.

Determine the power at \(\theta=0.5\)

The power of a hypothesis test is the probability that it correctly rejects the null hypothesis when the alternative hypothesis is true. It is denoted as \(1 - \beta\). Considering our alternative hypothesis is \(H_1: \theta = 0.5\), we can find the power of the test as \(1 - \beta = P(C | H_1)\). This equals to \(P(\sum_{1}^{10} x_{i} \geq 3 | \theta = 0.5)\), which can be obtained using the Poisson distribution's probability mass function.

Key concepts

Null Hypothesis

The null hypothesis, often denoted as \(H_0\), is a statement that there is no effect or no difference, and it represents the status quo or a baseline for comparison in hypothesis testing. In the context of the Poisson distribution, the null hypothesis posits a specific value for the average number of events within a given interval, \(\theta\).

In our exercise, the null hypothesis claims that \(\theta = 0.1\), implying that on average, there is 0.1 occurrence of a certain event per observation or time period. The idea behind setting a null hypothesis is to provide a benchmark that the observed data can be compared against, to determine whether there is enough evidence to suggest a significant deviation from what is expected under the null hypothesis.

Critical Region

The critical region, sometimes referred to as the rejection region, is the set of all outcomes of a statistical test that leads to the rejection of the null hypothesis. It is determined based on the desired significance level of the test and the distribution of the test statistic.

In the exercise provided, the critical region is defined by the sum of the random sample values being greater than or equal to 3, expressed mathematically as \(\sum_{1}^{10} x_{i} \geq 3\). This critical region is 'best' in the sense that it optimizes some criterion, such as minimizing the probability of a type II error or maximizing power for a given significance level.

Significance Level

The significance level, denoted \(\alpha\), is the threshold at which we reject the null hypothesis. In simpler terms, it is the probability of making a type I error, where we incorrectly reject the null hypothesis when it is actually true.

The significance level is chosen by the researcher before conducting the test and is commonly set at 0.01, 0.05, or 0.10. In the provided solution, the significance level is calculated by finding the probability that the sum of the Poisson distributed random variables exceeds the threshold defined by the critical region, given the null hypothesis is true.

Power of a Test

The power of a test, often understood as \(1 - \beta\), is the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true. It measures the test's ability to detect an effect if there is one.

A high power is desirable as it means there's a lower chance of committing a type II error, where we fail to reject the null hypothesis even though the alternative is true. In our exercise, the power is determined at the value of \(\theta\) specified in the alternative hypothesis, \(H_1: \theta = 0.5\), providing us with the likelihood that our test will identify the difference when the true mean is indeed 0.5.

Probability Mass Function

The probability mass function (PMF) describes the probability distribution of a discrete random variable. It's a function that gives the probability that a discrete random variable is exactly equal to some value. We use the PMF to calculate probabilities involving discrete data, as opposed to a probability density function (PDF) which is used for continuous data.

In our Poisson distribution exercise, the PMF is utilized to calculate both the significance level and the power of the test by estimating the likelihood of observing certain sums of the sample data, when following a Poisson distribution with a specified mean \(\theta\).