Applied Statistics Using SPSS, STATISTICA, MATLAB and R

A.7 The Binomial and Normal Distributions 421
Gaussian) density and is represented in Figure A.7 together with the distribution
function, also known as error function. Notice that, taking into account the
properties of the mean and variance, this new random variable has zero mean and
unit variance.
The approximation between normal and binomial distributions is quite good
even for not too large values of n. Figure A.6 shows the situation with n = 50,
p = 0.5. The maximum deviation between binomial and normal approximation
occurs at the middle of the distribution and is 0.056. For n = 1000, the deviation is
0.013. In practice, when np or nq are larger than 25, it is reasonably safe to use the
normal approximation of the binomial distribution.
Note that:
X n − np
X n
Z = ~ N 0 , 1 ⇒ Pˆ
= ~ N , A. 39
p,
pq / n
npq
n
where Nµ, σ is the Gaussian distribution with mean µ and standard deviation σ, and
the following density function:
1
2 2
−(
x−µ
) / 2σ
f ( x)
= e . A. 40
2π
σ
Both binomial and normal distribution values are listed in tables (see Appendix
D) and can also be obtained from software tools (such as EXCEL, SPSS,
STATISTICA, MATLAB and R).
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
F (x )
0
x
0 5 10 15 20 25 30 35 40 45 50
Figure A.6. Normal approximation (solid line) of the binomial distribution (grey
bars) for n = 50, p = 0.5.
Example A. 18
Q: Compute the tolerance of the previous Example A.17 using the normal
approximation.
A: Like before, we consider the worst-case situation with p = q = ½. Since
σ = 1/
4n
= 0.01, and the 95% confidence level corresponds to the interval
pˆ