When X1, X2, …, Xn are independent Poisson random

Chapter 9, Problem 173MEE

(choose chapter or problem)

When \(X_{1}, X_{2}, \ldots, X_{n}\) are independent Poisson random variables, each with parameter \(\lambda\), and n is large, the sample mean \(\bar{X}\) has an approximate normal distribution with mean \(\lambda\) and variance \(\lambda / n\). Therefore,

\(Z=\frac{\bar{X}-\lambda}{\sqrt{\lambda / n}}\)

has approximately a standard normal distribution. Thus, we can test \(H_{0}: \lambda=\lambda_{0}\) by replacing \(\lambda\) in Z by \(\lambda_{0}\) . When \(X_{i}\) are Poisson variables, this test is preferable to the large-sample test of Section 9-2.3, which would use \(S / \sqrt{n}\) in the denominator because it is designed just for the Poisson distribution. Suppose that the number of open circuits on a semiconductor wafer has a Poisson distribution. Test data for 500 wafers indicate a total of 1038 opens. Using \(\alpha=0.05\), does this suggest that the mean number of open circuits per wafer exceeds 2.0?