# a problem on normal approximation to the poisson - from Durrett's book

I came across a problem from Durrett's book Elementary Probability

P.217 Example 6.26

Normal approximation to the Poisson. Each year inMythica, an average of 25 letter carriers are bitten by dogs. In the past year, 33 incidents were reported. Is this number exceptionally high?

Solution: Assuming that dog bites are a rare event, we use the Poisson distribution for the number of dog bites. As we observed in Example 6.3, a Poisson with mean 25 is the sum of 25 independent Poisson mean 1 random variables, so we can use the normal to approximate the Poisson.

The mean is $25$, while the standard deviation is $\sqrt{25} = 5$.Writing the observed event as $≥32.5$ we see that this is $(32.5−25)/5 = 1.5$ standard deviations above the mean, so the normal approximation is $$P(\chi ≥ 1.25) = 1 − 0.8943 = 0.1057$$so this is not a very unusual event.

Why it is $P(\chi ≥ 1.25)$, not $P(\chi ≥ 1.5)$? Can anybody explain it to me?

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But $P(\chi ≥ 1.5) = 1- 0.9332 = 0.0668$, so it's less likely to be just typos – xoofee Nov 18 '12 at 5:47