To determine which sample would result in a better representation of data using a binomial distribution model, we should consider the sample size relative to the population size. A good rule of thumb for using a binomial distribution effectively is that the sample size should be large enough to provide a reliable approximation of the distribution, typically involving more than a few trials.
Out of the provided options, the sample sizes and populations can be evaluated as follows:
- a sample of 1,000 people to represent a population of 11,000 - This is a large sample relative to the population, providing a good representation.
- a sample of 300 people to represent a population of 700 - This is also a good sample size, as it covers a significant proportion of the population.
- a sample of 500 people to represent a population of 800 - This is a very high sample size relative to the population size, which could lead to effective representation but may have some diminishing returns due to the high proportion.
- a sample of 100 people to represent a population of 200 - This is a smaller sample relative to the population, but it still constitutes a significant fraction of the population.
Among these options, the sample of 300 people to represent a population of 700 or the sample of 500 people to represent a population of 800 would result in strong representations. However, the sample of 500 people to represent a population of 800 has an even higher proportion (62.5%), potentially leading to a better fit for the binomial distribution model since the results will be less variable and more stable given the larger sample relative to its population.
So, the best option is:
a sample of 500 people to represent a population of 800.
1 answer