Large Sample Test for Proportion
A random sample of size n (n > 30) has a sample proportion p of members possessing a certain attribute (success). To test the hypothesis that the proportion p in the population has a specified value p_{0}.
The Null Hypothesis is H_{0}: p = p_{0}.
The Alternative Hypothesis is (i) H_{1}: p ≠ p_{0 }or (ii) H_{1}: p < p_{0}, or (iii) H_{1}: p > p_{0}.
Since n is large, the sampling distribution of is approximately normal.
If H_{0} is true, the test statistic z = is approximately normally distributed.
The critical region for z depending on the nature of H_{1} and level of significance α is given in the following table:
Rejection Rule for H_{0}: p = p_{0}
Level of significance
10%
5%
1%
Critical region for p p_{0}
| z | > 1.64
| z | > 1.96
| z | > 2.58
Critical region for p < p_{0}
z < -1.28
z < -1.64
z < -2.33
Critical region for p > p_{0}
z > 1.28
z > 1.64
z > 2.33