251proport 4/28/06
For large samples we can use the Normal approximation to the binomial distribution. Use p for the population probability of success and q for the population probability of failure, and let p
 x n
and q

1
 p be the corresponding sample quantities. Remember that a proportion is always between zero and one. For the interval write p
 p
 z

2 s p
, where s p
 p q
. An example follows. n
Exercise 8.24 [Exercise 8.22 in 9 th edition] (Exercise 8.22 in 8 th edition): The problem asks for a 95% confidence interval for the population proportion of successes when n

200 and x

50
Solution: Since 1
  
.
95 ,
 
.
05 and z
 z

2
 z
.
05

1 .
960 . p
 x n

50
200

.
250 . q


10
.
250 p can say

1


1 .
960
P

.
190 x

1

.
250

.
750 .
So the confidence interval is p
 p
 z

2 s p
 p
 z

2 p q n n
.
250

.
750

 p
200

.
310



.
250
.
95

1 .
960 .
0009375

.
250

.
060 or .
190
 p

.
310 .
More formally, we
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 hits zero at .
250

3 .
90 .
0009375 , this is a good part of the area under the curve.