Kye

2021-02-09

Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither?
(a)$n=40$,$p=.05$

Elberte

The Rule of five:
The normal with means np and npq can be used to approximate the binomial distribution with parameters n and pif $npq>5$.
Here, $n=40$, $p=0.05$.
$npq=40×0.05×0.95=19<5$
Normal approximation cannot be used for the binomial distribution with $n=40$, $p=0.05$.
The direct approximation of the binomial by Poisson says that the binomial distribution with parameters n and p has the same distribution as the Poisson with the parameter np when npis large.
The value of np is obtained as shown below:
$np=40×0.05=2$
Since the value of n is large and pis small and the value of np is small, the binomial distribution can be approximated to Poisson.
That is, the binomial distribution with $n=40$, $p=0.05$ cannot be approximated to a normal distribution but can be approximated to a Poisson distribution.

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