Sampling with Replacement

Definition

Sampling with replacement is a statistical technique where items are selected from a population, and after each selection, the item is returned to the population before the next selection is made. This means that the same item can be selected multiple times during the sampling process, allowing for the possibility of repetition in the sample.

5 Must Know Facts For Your Next Test

1. Sampling with replacement is often used in simulations and statistical modeling to generate random samples that can be used to estimate population parameters.
2. The probability of selecting an item in a sampling with replacement process remains constant for each selection, as the item is returned to the population before the next selection is made.
3. Sampling with replacement can lead to the same item being selected multiple times in a single sample, which can affect the variance and standard error of the sample statistics.
4. Sampling with replacement is a useful technique when the population is large, and the researcher is interested in understanding the distribution of a statistic rather than the precise value of the parameter.
5. Sampling with replacement is a key concept in the study of Bernoulli trials and other probability-based statistical models.

Review Questions

• Explain how sampling with replacement differs from sampling without replacement in the context of independent and mutually exclusive events.
  • In the context of independent and mutually exclusive events, sampling with replacement means that the selection of an item from the population does not affect the probability of selecting any other item, as the selected item is returned to the population before the next selection is made. This is in contrast to sampling without replacement, where the selection of an item reduces the probability of selecting that same item in subsequent trials, as it is no longer available in the population. The independence and mutual exclusivity of events are maintained in sampling with replacement, as each selection is a separate and independent event with a constant probability of occurrence.
• Describe how sampling with replacement can impact the variance and standard error of sample statistics, and explain the implications for making inferences about the population.
  • Sampling with replacement can lead to the same item being selected multiple times in a single sample, which can affect the variance and standard error of the sample statistics. When an item is selected more than once, it contributes more to the sample mean and other statistics, potentially increasing the variance and standard error of the sample. This can have implications for making inferences about the population, as the sample statistics may not accurately represent the true population parameters. Researchers need to be aware of this potential impact and adjust their statistical analyses accordingly, particularly when the sample size is small or the population is not well-known.
• Analyze the role of sampling with replacement in the study of Bernoulli trials and other probability-based statistical models, and explain how it relates to the concept of independent and mutually exclusive events.
  • Sampling with replacement is a key concept in the study of Bernoulli trials and other probability-based statistical models. In Bernoulli trials, each experiment has two possible outcomes (success or failure), and the probability of success remains constant across all trials. Sampling with replacement is essential in these models, as it ensures that the probability of success remains the same for each trial, regardless of the outcomes of previous trials. This is because the selected item (i.e., the outcome of the trial) is returned to the population before the next selection is made, maintaining the independence and mutual exclusivity of the events. The relationship between sampling with replacement and independent and mutually exclusive events is crucial in the development and analysis of probability-based statistical models, as it allows researchers to make accurate inferences about population parameters and the distribution of statistics.

Related terms

Sampling without Replacement: In this method, items are selected from a population, and once selected, they are not returned to the population before the next selection is made. This ensures that each item can only be selected once during the sampling process.

Probability Sampling: Probability sampling is a method where each item in the population has a known, non-zero chance of being selected for the sample. This includes techniques like simple random sampling, systematic sampling, and stratified sampling.

Bernoulli Trials: Bernoulli trials are a series of independent experiments, each with two possible outcomes (success or failure), where the probability of success remains constant across all trials.