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.

5 Must Know Facts For Your Next Test

1. Sampling with replacement is often used in simulations and statistical modeling to generate random samples from a population.
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 after each selection.
3. Sampling with replacement can be used to estimate population parameters, such as the mean or variance, by repeatedly drawing samples from the population.
4. The probability of selecting a particular item in a sampling with replacement process follows a geometric distribution, where the probability of selecting the item remains the same for each selection.
5. Sampling with replacement is a useful technique when the population size is large, and the researcher is interested in the overall distribution of the population rather than the specific items selected.

Review Questions

• Explain how sampling with replacement differs from sampling without replacement, and provide an example of when each technique might be used.
  • Sampling with replacement allows for the same item to be selected multiple times during the sampling process, while sampling without replacement ensures that each item can only be selected once. Sampling with replacement is often used in simulations and statistical modeling, where the researcher is interested in the overall distribution of the population. For example, in a study of the heights of people in a city, sampling with replacement could be used to generate random samples and estimate the mean and standard deviation of the population. On the other hand, sampling without replacement might be used in a survey where the researcher wants to ensure that each individual in the population is only counted once.
• Describe how the probability distribution of the selected items in a sampling with replacement process differs from a sampling without replacement process.
  • In a sampling with replacement process, the probability of selecting a particular item remains constant for each selection, as the item is returned to the population after each selection. This means that the probability distribution of the selected items follows a geometric distribution, where the probability of selecting a specific item is the same for each selection. In contrast, in a sampling without replacement process, the probability of selecting a particular item changes with each selection, as the item is not returned to the population. This means that the probability distribution of the selected items follows a hypergeometric distribution, which takes into account the changing probabilities as items are removed from the population.
• Analyze the advantages and disadvantages of using sampling with replacement in the context of statistical modeling and estimation.
  • One of the main advantages of using sampling with replacement in statistical modeling and estimation is that it allows for the generation of random samples from a population, which can be used to estimate population parameters, such as the mean or variance. This can be particularly useful when the population size is large, and the researcher is interested in the overall distribution of the population rather than the specific items selected. Additionally, sampling with replacement can be computationally efficient, as it does not require keeping track of which items have been selected. However, a potential disadvantage of sampling with replacement is that it may not accurately reflect the true distribution of the population, especially if the population is small or the items are highly correlated. In such cases, sampling without replacement may be more appropriate, as it better captures the underlying structure of the population.

Related terms

Sampling without Replacement: Sampling without replacement is a statistical technique where items are selected from a population, and once selected, they are not returned to the population before the next selection is made. This means that each item can only be selected once during the sampling process.

Probability Distribution: A probability distribution is a mathematical function that describes the likelihood of different possible outcomes in a given situation or experiment.

Binomial Distribution: The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent trials, where each trial has only two possible outcomes, such as success or failure.