Question

Let \(X\) have a Poisson distribution with mean \(\theta\). Consider the simple hypothesis \(H_{0}: \theta=\frac{1}{2}\) and the alternative composite hypothesis \(H_{1}: \theta<\frac{1}{2} .\) Thus \(\Omega=\left\\{\theta: 0<\theta \leq \frac{1}{2}\right\\}\). Let \(X_{1}, \ldots, X_{12}\) denote a random sample of size 12 from this distribution. We reject \(H_{0}\) if and only if the observed value of \(Y=X_{1}+\cdots+X_{12} \leq 2\). Show that the following \(\mathrm{R}\) code graphs the power function of this test: theta=seq \((.1, .5, .05) ;\) gam=ppois \((2\), theta*12 \()\) plot (gam "theta, pch=" ", xlab=expression(theta), ylab=expression(gamma)) lines (gam "theta) Run the code. Determine the significance level from the plot.

Short answer

The implementation of the R code gives the graph of the power function for the hypothesis test, and the significance level is determined from the plot as the value of gamma corresponding to \(\theta = \frac{1}{2}\).

Step-by-step solution

Understand the problem

We are given that \(X\) follows a Poisson distribution with mean \(\theta\). We have a null hypothesis, \(H_0: \theta = \frac{1}{2}\), and an alternative hypothesis, \(H_1: \theta < \frac{1}{2}\). A random sample of size 12 is drawn from this distribution. The null Hypothesis \(H_0\) is rejected if the observed value of \(Y = X_1 + X_2 + \ldots + X_{12} \leq 2\).

Understand and Run the R code

The provided code in R is used to compute the power function for this statistical test. The first line of the code creates a sequence of hypothetical values for \(\theta\) ranging from 0.1 to 0.5 in increments of 0.05. The second line of the code calculates the power function (gamma) for each hypothetical value of \(\theta\) using the ppois function, which calculates the cumulative distribution function (CDF) of the Poisson distribution. The third and fourth lines of the R code plot the power function against the hypothetical values of \(\theta\).

Interpret the plot and determine the significance level

After running the R code and plotting the graph, analyze the graph to determine the significance level. The significance level is the \(gam\) value when \(\theta=\frac{1}{2}\), which is the null hypothesis. In the plotted graph, find the \(gam\) value corresponding to \(\theta=\frac{1}{2}\), since this is the probability of rejecting the null hypothesis when it is true (Type I error).

Key concepts

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known average rate and independently of the time since the last event. For instance, it can be used to model the number of emails an individual receives per hour or the number of stars that appear in a certain piece of the sky.

To define a Poisson distribution, you need to know the average rate of occurrence, denoted as \( \lambda \), or in the context of the provided exercise, \( \theta \). The probability of observing exactly \( k \) events in an interval can be found with the formula:\[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]where \( e \) is the base of the natural logarithm, and \( k! \) is the factorial of \( k \).

Our exercise deals with a scenario where the mean \( \theta \) of the Poisson distribution is hypothesized to be \( \frac{1}{2} \). We are using the sum of twelve random variables drawn from this distribution to test our hypothesis.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (\(H_0\)) is a statement made about a population parameter that we aim to test for validity. It commonly represents the idea of 'no effect' or 'no difference.' The alternative hypothesis (\(H_1\)), on the other hand, is what researchers suspect might be true instead of the null hypothesis.

In our specific case, the null hypothesis is \(H_0: \theta = \frac{1}{2}\), suggesting that the mean event rate of the Poisson distribution is 0.5. The alternative hypothesis is \(H_1: \theta < \frac{1}{2}\), indicating that the mean event rate might be less than 0.5. The conclusion from the test will either support the null hypothesis or suggest that the alternative hypothesis may be true, prompting further research.

To reject the null hypothesis in our scenario, we need the observed value \(Y = X_1 + X_2 + \ldots + X_{12}\) to be less than or equal to 2. This decision criterion is chosen based on knowledge about the behavior of the Poisson distribution and the desired significance level of the test.

Significance Level

The significance level, denoted by \(\alpha\), is a threshold chosen by the researcher that defines how stringent the hypothesis test is. It represents the probability of rejecting the null hypothesis when it is actually true, a Type I error. Common levels of significance are 0.05, 0.01, and 0.10.

In our textbook problem, we determine the significance level by running the R code and observing the resulting plot. The significance level corresponds to the probability associated with the critical value, the point beyond which we reject \(H_0\). This is calculated for our specified value of \(\theta = \frac{1}{2}\) using the cumulative distribution function. By finding the gamma (the power function) value at our null hypothesis' \(\theta\), we identify the test's significance level, which represents our willingness to accept the risk of a Type I error. This plot essentially gives us the capability of the test to reject the null hypothesis for various values of \(\theta\) within the alternative hypothesis range.

R Statistical Software

R is a powerful software and programming language used for statistical computing and graphics. It allows users to perform a variety of statistical analyses, with capabilities ranging from simple calculations to complex predictive modeling. R is widely used by statisticians and data analysts for developing statistical software and analyzing data.

In our exercise, R is employed to calculate the power function for different values of \(\theta\), which helps to visualize the probability of correctly rejecting the null hypothesis when it is false. The commands mentioned in the exercise, such as \('seq'\),\('ppois'\), \('plot'\), and \('lines'\), are used to generate a sequence of values, evaluate the Poisson distribution's cumulative distribution function, and plot the results accordingly.

Running the provided R code contributes a graphical representation of the hypothesis test's strength and how it varies with different mean rates \(\theta\), allowing for a clear interpretation of the significance level and an understanding of the test's sensitivity to detect differences from the null hypothesis.