3 Some previous studies have shown a relationship between em

3. Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. a) The hospital is planning a new emergency-room facility. It wants enough beds in the emergency ward so that for at least 95% of normal-pollution days it will not need to turn anyone away. What is the smallest number of beds it should have to satisfy this criterion? b) The hospital also finds that on high-pollution days the number of admissions is Poisson-distributed with mean = 4.0 admissions per day. Answer problem a for high-pollution days. c) On a random day during the year, what is the probability there will be 4 admissions to the emergency ward, assuming there are 345 normal-pollution days and 20 high-pollution days? d) Answer Problem (a) for a random day during the year.

use excel commands

Solution

her e mean =2; number of admissions follows poison distribution

a) we need number of admissions , such that atleast 95% days need not turn anyone away

i.e P(X<=x) >=0.95

From poison tables P(X<=5)=0.9834 ; P(X<=4) =0.9473

So smallest number of beds = 5

b) when mean =4

P(X<=8)=0.9786 ; P(X<=7) =0.9489

so smallest number of beds =8

c) On a random day during the year, the probability there will be 4 admissions to the emergency ward, assuming there are 345 normal-pollution days and 20 high-pollution days = 20/365 = 0.055

d) Answer Problem (a) for a random day during the year. P(x=4) when mean =2 is

P(X=4) =0.09