🚀 go-pugleaf

In article <1992Oct8.202310.10461@ultb.isc.rit.edu>, jsvrc@rc.rit.edu (J A Stephen Viggiano) writes:
> 
> For the gaussian approximation, we were taught that the tails of the
> approximating distribution should be well within the limits of 0 and n. Thus,
> np - 5 (npq)**0.5 should be greater than zero, and np + 5(npq)**0.5 should
> be less than n, where q = 1 - p. In other words, the mean should be at least
> five standard deviations from both zero and n. Again, this is a rule of
> thumb. For p = 0.5 (the best case), n should be at least 25. Smaller or
> larger values of p will require a larger n to reduce the coefficient of
> variation.
> 
It is interesting to see the diversity of opinions in the text books on this.
Ott and Mendenhall even give two different rules in the same book! At
on place they say we need np > 5 and n(1-p) > 5 and elsewhere they say
np^ > 10 and n(1-p^) > 10. The first rule seems to be for calculating
binomial probabilities with p known, and the latter for inference when
p is unknown. They do not draw the student's attention to the apparent
discrepancy.
Larsen and Marx give a rule more like the one quoted above, except they
are far less conservative. They only require
np^ - 2 (np^q^)**0.5 > 0, and np^ + 2 (np^q^)**0.5 < n.
O & M's second suggestion is equivalent to
n > max(10/p^, 10/q^)
while L & M's is
n > max(a^2*p^/(1-p^), a^2*q^/(1-q^)) where a = 2.
Taking different values for a gives a host of possible rules and leaves
scope for adjusting a to suit the size of confidence interval we wish
to use. Taking a = 5 gives roughly the n > max(5/p, 5/q) rule.

Terry Moore