I have a doubt regarding when to approximate binomial distribution with Poisson distribution and when to do the same with Normal distribution.
It is my understanding that, when p is close to 0.5, that is binomial is fairly symmetric, then Normal approximation gives a good answer. However, when p is very small (close to 0) or very large (close to 1), then the Poisson distribution best approximates the Binomial distribution.
Also, when n is large enough to compensate, normal will work as a good approximation even when n is not close to 0.5 (n will work fine, but still Poisson will be better? )
However,consider the following question-
The probability of any given policy in a portfolio of term assurance policies lapsing before it expires is considered to be 0.15. For a group of 100 such policies, calculate the approximate probability that more than 20 will lapse before they expire.
Here n is 100 and p is 0.15 (which is not close to 0.5). In this case, the exact answer is 0.0663. The normal approximated answer is 0.06178 and the Poisson approximated answer is 0.08297.
My doubt is that, since p is closer to zero than it is to 0.5, shouldn't the Poisson approximation yield a better answer?
Here is a pmf plot I was able to create in MATLAB—looks like the normal (Gaussian) is pretty close, where as the Poisson misses the peak and has a fatter long tail.
Furthermore, looking at wiki (not always infallible!), according to NIST/SEMATECH, "22.214.171.124. Counts Control Charts", e-Handbook of Statistical Methods., Poisson is a good approximation for $p < 0.05$ (not 0.5) for $n > 20$.