Improperly handled or undercooked poultry and eggs are the most frequent cause o
ID: 2923466 • Letter: I
Question
Improperly handled or undercooked poultry and eggs are the most frequent cause of salmonella food poisoning. Salmonellosis can be deadly, although most individuals recover on their own. Cases must be reported to the CDC, where they are closely monitored. The CDC reports that the average number of salmonellosis cases per month in Wisconsin is 15.58. (a)If X is the monthly count of salmonellosis cases in Wisconsin, what distribution does X follow, approximately? Give the mean and standard deviation of X. (b)Use technology to find the probabilities P(X = 0), P(X 5), P(X 15), and P(X 25). What is the probability that there would be more than 25 cases of salmonellosis in Wisconsin in a given month? (c)As with South Dakota, September 1994 had an unusually high number of salmonellosis cases in Wisconsin with 48 cases that month. What is the probability that 48 or more cases would arise in one month in Wisconsin? What do you make of the fact that a salmonellosis outbreak occurred in two states at the same time? In fact, the salmonellosis outbreak was traced back to nationally distributed ice cream products from one factory. The factory was shut down until cleared, and its contaminated production recalled, thus preventing many more salmonellosis cases.
Explanation / Answer
The CDC reports that the average number of salmonellosis cases per month in Wisconsin is 15.58
(a)If X is the monthly count of salmonellosis cases in Wisconsin
pmf of P.D is = f ( k ) = e- x / x!
where
= parameter of the distribution.
x = is the number of independent trials
I.
the mean and the variance of the Poisson distribution are both equal to
mean = = 5
s.d = sqrt(5) = 2.2360
P( X = 0 ) = e ^-15.58 * 15.58^0 / 0! = 0.000000171274
P( X < = 5) = P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= e^-15.58 * 15.58 ^ 5 / 5! + e^-15.58 * 0 ^ 4 / 4! + e^-15.58 * ^ 3 / 3! + e^-15.58 * ^ 2 / 2! + e^-15.58 * ^ 1 / 1! + e^-15.58 * ^ 0 / 0!
= 0.00186,
P( X < = 15) = P(X=15) + P(X=14) + P(X=13) + P(X=12) + P(X=11) + P(X=10) + P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) +
= e^-15.58 * 15.58 ^ 15 / 15! + e^-15.58 * 0 ^ 14 / 14! + e^-15.58 * ^ 13 / 13! + e^-15.58 * ^ 12 / 12! + e^-15.58 * ^ 11 / 11! + e^-15.58 * ^ 10 / 10! + e^-15.58 * ^ 9 / 9! + e^-15.58 * ^ 8 / 8! + e^-15.58 * ^ 7 / 7!
e^-15.58 * ^ 6 / 6! + e^-15.58 * ^ 5 / 5! = 0.50889
P( X < = 25) = P(X=25) + P(X=24) + P(X=23) + P(X=22) + P(X=21) + P(X=20) + P(X=19) + P(X=18) + P(X=17) + P(X=16) + P(X=15) + .....P(X=0)
= e^-15.58 * 15.58 ^ 25 / 25! + e^-15.58 * 0 ^ 24 / 24! + e^-15.58 * ^ 23 / 23! + e^-15.58 * ^ 22 / 22! + e^-15.58 * ^ 21 / 21! + e^-15.58 * ^ 20 / 20! + e^-15.58 * ^ 19 / 19! + e^-15.58 * ^ 18 / 18! + e^-15.58 * ^ 17 / 17! + ........+ e ^-15.58 * 15.58^0 / 0!
e^-15.58 * ^ 16 / 16! + e^-15.58 * ^ 15 / 15! + = 0.99031
P( X > 25) = 1 -P ( X <= 25) = 1 - 0.9903 = 0.00969