Introduction to Probability and Statistics for Engineers and Scientists

210 Chapter 6: Distributions of Sampling Statistics

0.30

0.25

0.20

0.15

0.10

0.05

0.0024 6 810

x

(10, 0.7) 0.20

0.15

0.10

0.05

0.0 0 5 10 15 20

x

(20, 0.7)

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.0 0 5 10 15 20 25

x

(30, 0.7) 0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.0 010203040

x

(50, 0.7)

30 50

FIGURE 6.3 Binomial probability mass functions converging to the normal density.

variable with parametersn=450 andp=.3. Since the binomial is a discrete and the
normal a continuous distribution, it is best to computeP{X=i}asP{i−.5<X<i+.5}
when applying the normal approximation (this is called the continuity correction). This
yields the approximation

P{X>150.5}=P

{

X−(450)(.3)

√

450(.3)(.7)

≥

150.5−(450)(.3)

√

450(.3)(.7)

}

≈P{Z>1.59}=.06

Hence, only 6 percent of the time do more than 150 of the first 450 accepted actually
attend. ■

It should be noted that we now have two possible approximations to binomial proba-
bilities: The Poisson approximation, which yields a good approximation whennis large
andpsmall, and the normal approximation, which can be shown to be quite good when
np(1−p) is large. [The normal approximation will, in general, be quite good for values
ofnsatisfyingnp(1−p)≥10.]

6.3.1 Approximate Distribution of the Sample Mean

LetX 1 ,...,Xnbe a sample from a population having meanμand varianceσ^2. The central
limit theorem can be used to approximate the distribution of the sample mean

X=

∑n

i= 1

Xi/n