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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 p0.
The Null Hypothesis is H0: p = p0.
The Alternative Hypothesis is (i) H1: p ≠ p0 or (ii) H1: p < p0, or (iii) H1: p > p0.
Since n is large, the sampling distribution of is approximately normal.
If H0 is true, the test statistic z = is approximately normally distributed.
The critical region for z depending on the nature of H1 and level of significance α is given in the following table:
Rejection Rule for H0: p = p0
Level of significance
10%
5%
1%
Critical region for p p0
| z | > 1.64
| z | > 1.96
| z | > 2.58
Critical region for p < p0
z < -1.28
z < -1.64
z < -2.33
Critical region for p > p0
z > 1.28
z > 1.64
z > 2.33
Empirical Mode Where mode is ill-defined, its value may be ascertained by the following formula based upon the empirical relationship between Mean, Median and Mode: Mode = 3
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