When X₁, X₂, €¦, X_nare independent Poisson random variables, each with parameter λ, and n is large, the sample mean X has an approximate normal distribution with mean λ and variance λ/n. Therefore,

has approximately a standard normal distribution. Thus, we can test H₀: λ = λ₀ by replacing λ in Z by λ₀. When X_i are Poisson variables, this test is preferable to the large-sample test of Section 9-2.3, which would use S / ˆšn 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 α = 0.05, does this suggest that the mean number of open circuits per wafer exceeds 2.0?

X-1 Va /n