Normal approximation

When the sample size, n, is large, a normal distribution may be used as an approximation to the binomial.

Approximate p-value

We again test the hypotheses

H0 :   π  =  π0
HA :   π  <  π0

If n is large, the approximate normal distribution for x can be used to obtain the p-value for the test.

Home-based businesses owned by women

A study found that 369 out of 899 sampled home-based businesses were owned by women. Are they less likely to be owned by females than by males? The hypotheses are...

H0 :   π  =  0.5
HA :   π  <  0.5

where π = P(owned by female).

with a standard normal distribution. Since x is discrete,

P(X ≤ 369)   =  P(X ≤ 369.5)   =   P(X ≤ 369.9)   =   ...

To find this tail probability, any value of x between 369 and 370 might have been used when evaluating the z-score. The p-value can be more accurately estimate by using 369.5. This is called a continuity correction.

The continuity correction involves either adding or subtracting 0.5 from the observed count, x, before finding the z-score.