Question

data from 14 cities were combined for a 20 - year period, and the 280 city - years included a total of 137 homicides. also, finding the mean number of homicides per city - year. then 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
actual results 172 83 22 2 1
a. ( p(0)=0.6131 )
(round to four decimal places as needed)
b. ( p(1)=0.3 )
(round to four decimal places as needed)
c. ( p(2)=square )
(round to four decimal places as needed)

Explanation:

Step1: Calculate the mean

The total number of city - years \(n = 280\), and the total number of homicides \(x=137\). The mean \(\lambda=\frac{x}{n}=\frac{137}{280}\approx0.4893\)

Step2: Use the Poisson probability formula

The Poisson probability formula is \(P(k)=\frac{e^{-\lambda}\lambda^{k}}{k!}\), where \(k = 2\), \(\lambda\approx0.4893\), \(e\approx2.7183\)

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First, calculate \(2.7183^{-0.4893}=e^{-0.4893}\approx0.6076\)

Then \(0.6076\times0.2394 = 0.1455\)

Finally, \(\frac{0.1455}{2}=0.0728\)

Answer:

\(0.0728\)