The Problem
We have a single sample of n individuals; each individual either ‘possesses’ a characteristic of interest (e.g. is male, is pregnant, has died) or does not possess that characteristic (e.g. is female, is not pregnant, is still alive). A useful summary of the data is provided by the proportion of individuals with the characteristic. We are interested in determining whether the true proportion in the population of interest takes a particular value.
The Test of a Single Proportion
Assumptions
Our sample of individuals is selected from the population of interest. Each individual either has or does not have the particular characteristic.
Notation
r individuals in our sample of size n have the characteristic. The estimated proportion with the characteristic is p = r/n. The proportion of individuals with the characteristic in the population is π. We are interested in determining whether π takes a particular value, π1.
Rationale
The number of individuals with the characteristic follows the Binomial distribution (Chapter 8), but this can be approximated by the Normal distribution, providing np and n(1 − p) are each greater than 5.
Then p is approximately Normally distributed with:
an estimated mean = p and
Therefore, our test statistic, which is based on p, also follows the Normal distribution.
The 1/2n in the numerator is a continuity correction: it is included to make an allowance for the fact that we are approximating the discrete Binomial distribution by the continuous Normal distribution.
Refer z to Appendix A1.
Interpret the P-value and calculate a confidence interval for the true population proportion, π. The 95% confidence interval for π is approximated by
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