Let X represent the number of events that are observed to occur in n units of time or space, and assume X ∼ Poisson(nλ), where λ is the mean number of events that occur in one unit of time or space. Assume X is large, so that X ∼ N(nλ, nλ). Follow steps (a) through (d) to derive a level 100(1 − α)% confidence interval for λ. Then in part (e), you are asked to apply the result found in part (d).
a. Show that for a proportion 1 − α of all possible samples, X − zα/2σX < nλ < X + zα/2σX .
b. Let = X/n. Show that σ= σX /n.
c. Conclude that for a proportion 1−α of all possible samples,  − zα/2σ< λ <  + zα/2σ.
d. Use the fact that (( (/n to derive an expression for the level 100(1−α)% confidence interval for λ?
e. A 5 mL sample of a certain suspension is found to contain 300 particles. Let λ represent the mean number of particles per mL in the suspension. Find a 95% confidence interval for λ?