Data from 14 cities were combined for a 20-year period, and the total 280 city-years included a total of 151 homicides. After finding the mean number of homicides per city-year, find the probability that a randomly selected city-year has the following numbers of homicides, then compare the actual results to those expected by using the Poisson probabilities: Homicides each city-year? a. 0 b. 1 c. 2 d. 3 e. 4

a) P (0) = 0.5832

b) P (1) = 0.3145

c) P (2) = 0.0848

d) P (3) = 0.0153

e) P (4) = 0.0002

Step-by-step explanation:

Number of units in the data (n) = 280

No of homocides (k) = 151

Using Poisson distribution, probability can be calculated as

P (x) = (e^-α * α^x) / x!

α = k/n

α = 151/280

α = 0.5393

a) P (0) = (e^-0.5393 * 0.5393^0) / 0!

= (0.5832 * 1) / 1

= 0.5832

b) P (1) = (e^-0.5393 * 0.5393^1) / 1!

= (0.5832 * 0.5393) / 1

= 0.3145

c) P (2) = (e^-0.5393 * 0.5393^2) / 2!

= (0.5832 * 0.2908) / 2

= 0.1696/2

= 0.0848

d) P (3) = (e^-0.5393 * 0.5393^3) / 3!

= (0.5832 * 0.1569) / 6

= 0.0915/6

= 0.0153

e) P (4) = (e^-0.5393 * 0.5393^4) / 4!

= (0.5832 * 0.0846) / 24

= 0.0493/24

= 0.0002