Question

Will \(\hat{p}\) from a random sample from a population with \(60 \%\) successes tend to be closer to 0.6 for a sample size of \(n=400\) or a sample size of \(n=800 ?\) Provide an explanation for your choice.

Short answer

The sample proportion \(\hat{p}\) will tend to be closer to 0.6 for a sample size of \(n=800\) in accordance with the principle of the Law of Large Numbers.

Step-by-step solution

Understand the Principle

Understand the concept of the Law of Large Numbers which implies that as the sample size increases, sample statistics like the sample mean or sample proportion tend to get closer to their corresponding population parameters.

Apply the Principle

Apply this principle to the given problem. Here, we're working with sample proportions - \(\hat{p}\) - and a given population proportion (0.6).

Compare the Sample Sizes

Compare the two specified sample sizes, n=400 and n=800. Given the law of large numbers, the sample proportion from the larger sample size, n=800, will tend to be closer to the population proportion.