Question

To use a binomial distribution to approximate the count of successes in an SRS, why do we require that the sample size n be less than 10% of the population size N?

A) To satisfy the condition of having the same probability for each trial.

B) To satisfy the condition that the outcomes are binary.

C) To satisfy the condition of having a fixed number of trials.

D) To satisfy the condition of independence when sampling without replacement.

E) To ensure that the distribution is approximately Normal.

Answer 1

To maintain the independence of trials, which is necessary for a binomial distribution, the sample size n should be less than 10% of the population size N when sampling without replacement.

Step-by-step explanation:

To use a binomial distribution to approximate the count of successes in a simple random sample (SRS), we require that the sample size n be less than 10% of the population size N to satisfy the condition of independence when sampling without replacement (option D).

This 10% condition is necessary because, as the size of the sample approaches the size of the population, the trials become less independent of each other, influencing the probability of success for each subsequent trial. When the sample size is less than 10% of the population, the change in the probability of success from one trial to the next is small enough to be considered negligible, maintaining the independence of the trials, which is a fundamental assumption of the binomial distribution.

It is important to note that while technology tools such as calculators and computer software now make it easier to calculate binomial probabilities for large values of n, understanding the 10% condition is still crucial when approximating the binomial distribution using theoretical principles.