Replacement Sampling

Definition

Replacement sampling is a statistical technique where observations are drawn from a population, and after each observation is made, the selected item is returned to the population before the next observation is drawn. This allows for the possibility of the same item being selected multiple times during the sampling process.

5 Must Know Facts For Your Next Test

1. Replacement sampling is often used in the context of the binomial distribution, where the probability of success remains constant across trials.
2. The binomial distribution can be used to model the number of successes in a fixed number of independent Bernoulli trials with replacement.
3. Replacement sampling allows for the possibility of the same item being selected multiple times, which can lead to a different probability distribution compared to sampling without replacement.
4. The variance of a binomial distribution with replacement is greater than the variance of a binomial distribution without replacement, as the trials are independent.
5. Replacement sampling is commonly used in simulations and experiments where the population size is large and the probability of selecting the same item multiple times is negligible.

Review Questions

• Explain how replacement sampling differs from sampling without replacement in the context of the binomial distribution.
  • In replacement sampling, the selected item is returned to the population before the next item is drawn, allowing for the possibility of the same item being selected multiple times. This is different from sampling without replacement, where the selected item is not returned to the population, and the probability of selecting any given item changes with each draw. The key difference is that in replacement sampling, the probability of success remains constant across trials, whereas in sampling without replacement, the probability of success changes as the population size decreases. This difference in the underlying probability model leads to different probability distributions, with replacement sampling resulting in a binomial distribution and sampling without replacement leading to a hypergeometric distribution.
• Describe the relationship between replacement sampling and the variance of a binomial distribution.
  • In the context of the binomial distribution, replacement sampling has a direct impact on the variance of the distribution. When sampling with replacement, the variance of the binomial distribution is greater than the variance of a binomial distribution with sampling without replacement. This is because in replacement sampling, the trials are independent, and the probability of success remains constant across trials. In contrast, when sampling without replacement, the probability of success changes as the population size decreases, leading to a lower variance. The increased variance in replacement sampling is due to the possibility of the same item being selected multiple times, which adds more variability to the distribution.
• Explain why replacement sampling is commonly used in simulations and experiments with large population sizes.
  • Replacement sampling is often preferred in simulations and experiments with large population sizes because the probability of selecting the same item multiple times is negligible. When the population size is large, the impact of removing an item from the population and not returning it is minimal, as the probability of selecting that item again is very low. By using replacement sampling, researchers can simplify the sampling process, as they do not need to keep track of which items have been previously selected. Additionally, replacement sampling allows for the use of the binomial distribution, which is a well-understood and widely-used probability model, making the analysis and interpretation of the results more straightforward.

Related terms

Sampling with Replacement:

A sampling method where the selected item is returned to the population before the next item is drawn, allowing the same item to be selected more than once.

Bernoulli Trials:

A sequence of independent experiments, each with two possible outcomes (success or failure), where the probability of success remains constant across trials.

Binomial Distribution:

A probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success.