"Which of the following binomial distributions can be..." was answered by our experts

Chesley

Answered question

2021-02-11

Which of the following binomial distributions can be well approximated by a normal distribution? A Poisson distribution? Both? Neither?
(с)

n = 500

p = .001

Answer & Explanation

d2saint0

Skilled2021-02-12Added 89 answers

The Rule of five:
The normal with means np and npq can be used to approximate the binomial distribution with parameters n and p if $n p q > 5$ .
Here, $n = 500$ , $p = 0.001$ .
$n p q = 500 \times 0.001 \times 0.999 = 0.4995 < 5$
Normal approximation cannot be used for the binomial distribution with $n = 500$ , $p = 0.001$ .
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 np is large.
The value of np is obtained as shown below:
$n p = 500 \times 0.001 = 0.5$
Since the value of n is large and p is small and the value of np is small, the binomial distribution can be approximated to Poisson.
That is, the binomial distribution with $n = 500$ , $p = 0.001$ cannot be approximated to a normal distribution and it can be approximated to a Poisson distribution.

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