The number of rescue calls received by a rescue squad in a city follows a Poisson distribution with 2.83 per day. The squad can handle at most four calls a day. What is the probability that the squad will be able to handle all the calls on a particular day?

UkusakazaL

UkusakazaL

Answered question

2021-07-31

The number of rescue calls received by a rescue squad in a city follows a Poisson distribution with 2.83 per day. The squad can handle at most four calls a day.
a. What is the probability that the squad will be able to handle all the calls on a particular day?
b. The squad wants to have at least 95% confidence of being able to handle all the calls received in a day. At least how many calls a day should the squad be prepared for?
c. Assuming that the squad can handle at most four calls a day, what is the largest value of that would yield 95% confidence that the squad can handle all calls?

Answer & Explanation

opatovaL

opatovaL

Beginner2021-08-03Added 1 answers

Step 1
Given Information
Number of Rescue calls received follows a Poisson Distribution with μ=2.83 per day
The squad can handle at most four calls a day
P.M.F for Poisson distribution is given by
P(X=x)=eμ×μxx!
Step 2
a) Probability that the squad will be able to handle all the calls on a particular day is
P(X4)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)
e2.83×2.8300!+e2.83×2.8311!+e2.83×2.8322!+e2.83×2.8333!+e2.83×2.8344!
=0.059+0.167+0.2229+0.1577
=0.8430
Step 3
b)
The squad wants to have at least 95% confidence of being able to handle all the calls received in a day. So, to find the number of calls to satisfy the above condition
Probability that the number of calls to have at least 95% confidence is
P(Xx)=x=0xe2.83×2.83xx!0.95
Taking x=1
P(X1)=0.226 (Using Excel = POISSON.DIST(1,2.83,TRUE))
Taking x=2
P(X2)=0.462 (Using Excel = POISSON.DIST(2,2.83,TRUE))
Taking x=3
P(X3)=0.685 (Using Excel = POISSON.DIST(3,2.83,TRUE))
Taking x=4
P(X4)=0.843 (Using Excel = POISSON.DIST(4,2.83,TRUE))
Taking x=5
P(X5)=0.932 (Using Excel = POISSON.DIST(5,2.83,TRUE))
Taking x=6
P(X6)=0.97 (Using Excel = POISSON.DIST(6,2.83,TRUE))
The number of calls to home at least 95% confidence is 6
Step 4
c)
Assume that the squad can handle at most four calls a day, then finding the largest value of µ that would yield 95% confidence that the squad can handle all calls.
P(X4)=x=0xe2.83×2.83xx!0.95
By taking different values of μ, the minimum value of μ to satisfy the above condition
P(X4)=x=0xe1.97×1.97xx!=0.95001
The minimum value of μ=1.97

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