Chapter 8, Problem 71SE

When X₁, X₂,…, X_n are independent Poisson variables, each with parameter μ. and n is large, the sample mean $\bar{X}$ has approximately a normal distribution with μ = E( $\bar{X}$) and V( $\bar{X}$) = μ/n. This implies that

$Z=\frac{\bar{X}-\mu }{\sqrt{\mu /n}}$

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 $S\text{/}\sqrt{n}$ (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, docs this suggest that the true average number of weekly requests exceeds 4.0? Test using α = .02.