A hospital has observed that the number of nosocomial (hospital-acquired) pneumonias

(NP) in its intensive care unit follows a Poisson distribution with a rate of 1.8 per month. The hospital’s infection-control officer monitors the number of NP each month, and calls for an expensive additional equipment sterilization effort if four or more infections are reported. The alternative hypothesis is that the rate of infections has increased.

(a) Assuming that the rate of infections has not increased, what is the probability that the officer will call for the sterilization effort?

(b) The officer repeats the monitoring every month. Over a 12-month period in which the rate stays at 1.8, what is the probability the officer will call for the sterilization effort at least once? Assume each month’s count of new infections is independent of the other months. LO.1