The requirements for constructing a confidence interval about p are satisfied as the sample is a simple random sample, np(1−p)=251.806 (greater than or equal to 10), and the sample size is less than or equal to 5% of the population size.
The satisfaction of specific requirements is crucial when constructing a confidence interval for a population proportion (p). In this case, the first condition ensures that the sample is a simple random sample, which means each individual in the population has an equal chance of being chosen. The second condition, np(1−p)=251.806, being greater than or equal to 10, is a prerequisite for using the normal approximation to the binomial distribution, making the interval estimation more reliable.
Lastly, the third condition indicates that the sample size is less than or equal to 5% of the population size, ensuring that the sample doesn't exert a substantial influence on the entire population. These three conditions collectively support the validity of constructing a confidence interval for the population proportion, providing a reasonable level of confidence in the accuracy of the interval estimate.