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.
Adverse reactions to drug
A pharmaceutical company claims that only 1% of a drug's users experience adverse reactions. An agency monitors 2500 patients taking the drug and observes adverse reactions in 37 cases. Is the occurrence of adverse reactions more common than claimed by the company?
H0 : π = 0.01
HA : π > 0.01
where π = P(adverse effect).
with a standard normal distribution. Since x is discrete,
P(X ≥ 37) = P(X ≤ 36.5) = P(X ≤ 36.9) = ...
To find this tail probability, any value of x between 36 and 37 might have been used when evaluating the z-score. The p-value can be more accurately estimate by using 36.5. This is called a continuity correction. This more accurate p-value in the above example is 0.011 leading to a similar conclusion.
The continuity correction involves either adding or subtracting 0.5 from the observed count, x, before finding the z-score.