Final answer:
To construct a confidence interval for a population proportion, a sample must meet specific criteria, including being a simple random sample, having np' and nq' greater than five, and the sample size not overstepping 5% of the population. The correct choice depends on the specifics of the sample data.
Step-by-step explanation:
To verify that the requirements for constructing a confidence interval about a population proportion p are satisfied, several criteria must be met:
- The sample must be a simple random sample.
- The sample size should not exceed 5% of the population to avoid over-sampling effects.
- The number of successes np' and the number of failures nq' in the sample should both be greater than five (np' ≥ 5 and nq' ≥ 5).
Without the full context of the sampling method, population size, and the observed number of successes and failures, we cannot definitively answer the given options. However, to answer the sample question specifically, if the number of successes np' and the number of failures nq' are both greater than five, then option A would be correct, as the minimum requirement for the normal approximation to the binomial distribution to construct the confidence interval is met.
If the sample is not a simple random sample, option B would be correct. If the sample size is more than 5% of the population, then option C applies. Lastly, if the product of the sample size and the sample proportion (and its complement) do not meet the condition np' × (1-p') > 10, then option D would be the correct answer.