Page 321 - Fundamentals of Probability and Statistics for Engineers

304 Fundamentals of Probability and Statistics for Engineers

and
2
P > 752:3†ˆ 0:95:

9.3.2.4 Confidence Interval for a Proportion

Consider now the construction of confidence intervals for p in the binomial
distribution

k
p X k†ˆ p 1 p† 1 k ; k ˆ 0; 1:
In the above, parameter p represents the proportion in a binomial experiment.
Given a sample of size n from population X, we see from Example 9.10 that an
unbiased and efficient estimator for p is X . For large n, random variable X is
approximately normal with mean p and variance p 1 p)/n.
Defining

p 1 p†
1=2
U ˆ X p† ; 9:146†
n
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
P u =2 < U < u =2 †ˆ 1 : 9:147†

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

" #
p 1 p†
1=2
P u =2 < X p† < u =2 ˆ 1 : 9:148†
n
In order to determine confidence limits for p, we need to solve for p satisfying
the equation

1=2

p 1 p†
jX pj u =2 ;
n
or, equivalently

u 2 =2 p 1 p†
2
X p† : 9:149†
n

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