Question

The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter

μ = 50.

(Round your answers to four decimal places.)

(a)

Calculate the approximate probability that between 35 and 70 tickets are given out on a particular day.

(b)

Calculate the approximate probability that the total number of tickets given out during a 5-day week is between 225and 285.

(c)

Use software to obtain the exact probabilities in (a) and (b) and compare to their approximations.

Calculate the exact probability that between 35 and 70 tickets are given out on a particular day.

Calculate the exact probability that the total number of tickets given out during a 5-day week is between 225 and 285.

Answer 1

Normal approximation
mean = 50
sd = sqrt(50) = 7.0711

a)
P(35 < X < 70)
= P((35 - 50)/7.0711 < z < (70 - 50)/7.0711)
= P(-2.1213 < z < 2.8284)
= 0.9807

b)
mean = 250
sd = sqrt(5*50) = 15.8114

P(225 < X < 285)
= P((225 - 250)/15.8114 < z < (285 - 250)/15.8114)
= P(-1.5811 < z < 2.2136)
= 0.9296

c)

P(35 < X < 70) = 0.9862
=POISSON.DIST(70,50,TRUE)-POISSON.DIST(34,50,TRUE)

P(225 < X < 285) = 0.9348
=POISSON.DIST(285,250,TRUE)-POISSON.DIST(224,250,TRUE)