251proport 1/14/08
A Confidence Interval for a Proportion in Large Samples.
For large samples we can use the Normal approximation to the binomial distribution. Use p for
x
and
n
q  1  p be the corresponding sample quantities. Remember that a proportion is always between zero and
the population probability of success and q for the population probability of failure, and let p 
one. For the interval write p  p  z s p , where s p 
2
pq
. An example follows.
n
Exercise 8.24 [Exercise 8.22 in 9th edition] (Exercise 8.22 in 8th edition): The problem asks for a 95%
confidence interval for the population proportion of successes when n  200 and x  50
x
50
Solution: Since 1    .95 ,   .05 and z  z  z.05  1.960 ttable. p  
 .250 .
2
n 200
q  1 p  1
x
 1  .250  .750 . So the confidence interval is p  p  z  2 s p  p  z  2
n
pq
n
.250 .750 
 .250  1.960 .0009375  .250  .060 or .190  p  .310. More formally, we
200
can say P.190  p  .310  .95 or make a diagram. The diagram would be a Normal curve with .250
at the center and with an area marked 95% extending from 0.190 to 0.310. Note that since the curve
 .250  1.960
hits zero at .250  3.90 .0009375 , this is a good part of the area under the curve.