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

Kye 2021-02-09 Answered
Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither?
(a)n=40,p=.05
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Expert Answer

Elberte
Answered 2021-02-10 Author has 95 answers

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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