Fundamentals of Probability and Statistics for Engineers

and

9.3.2.4 Confidence Interval for a Proportion

Consider now the construction of confiden ce intervals for p in the binomial
distribution

In the above, parameter p represents the proportion in a binomial exper iment.
Given a sample of size n from population X, we see from Example 9.10 that an
unbiased and efficient estimator for p isX. For large n, random variableXis
approximately normal with mean p and variance
Defining

random variable U tends to N(0,1) as n becomes large. In terms of U, we have
the same situation as in Section 9.3.2.1 and Equation (9.129) gives

The substitution of Equation (9.146) into Equation (9.147) gives

In order to determine co nfiden ce limits for p, we need to so lve for p satisfying
the equation

or, equivalently

304 Fundamentals of Probability and Statistics for Engineers

P…^2 > 752 : 3 †ˆ 0 : 95 :

pX…k†ˆpk… 1 p†^1 k; kˆ 0 ; 1 :

p1p)/n.

Uˆ…Xp†

p… 1 p†

n

 1 = 2

; … 9 : 146 †

P…u= 2 <U<u= 2 †ˆ 1 : … 9 : 147 †

Pu= 2 <…Xp†

p… 1 p†

n

 1 = 2

<u= 2

"

ˆ 1 : … 9 : 148 †

jXpj

p… 1 p†

n

 1 = 2

u= 2 ;

…Xp†^2 

u^2 = 2 p… 1 p†

n

: … 9 : 149 †