When X1, X2, … , Xnare independent Poisson variables, each with parameter μ, and n is la...

When X₁, X₂, … , X_nare independent Poisson variables, each with parameter μ, and n is large, the sample mean has approximately a normal distribution with and . This implies that

has approximately a standard normal distribution. For testing H₀: μ = μ₀, we can replace μ by μ₀ in the equation for Z to obtain a test statistic. This statistic is actually preferred to the large-sample statistic with denominator (when the X_i’s are Poisson) because it is tailored explicitly to the Poisson assumption. If the number of requests for consulting received by a certain statistician during a 5-day work week has a Poisson distribution and the total number of consulting requests during a 36-week period is 160, does this suggest that the true average number of weekly requests exceeds 4.0? Test using α = .02.