Sampling with Replacement

Definition

Sampling with replacement is a statistical technique where items are selected from a population and then returned to the population before the next item is selected. This means that the same item can be selected multiple times during the sampling process, allowing for the possibility of the same value appearing in the sample more than once.

5 Must Know Facts For Your Next Test

1. Sampling with replacement is a key assumption of the Central Limit Theorem, which states that the sampling distribution of the sample mean will be approximately normal as the sample size increases.
2. In the context of the Pocket Change example, sampling with replacement means that each time a coin is selected from a person's pocket, it is returned to the pocket before the next coin is selected.
3. Sampling with replacement allows for the possibility of the same value appearing multiple times in a sample, which can affect the sample variance and other statistical properties.
4. The Central Limit Theorem applies to sampling with replacement, as well as sampling without replacement, as long as the sample size is sufficiently large.
5. Sampling with replacement is often used in simulations and bootstrapping techniques to estimate the properties of a statistic, such as its standard error or confidence interval.

Review Questions

• Explain how sampling with replacement relates to the Central Limit Theorem in the context of the Pocket Change example.
  • In the Pocket Change example, the Central Limit Theorem states that the sampling distribution of the sample mean of the amount of pocket change will be approximately normal, regardless of the actual distribution of the amount of pocket change in the population. This is true even if the sampling is done with replacement, meaning that each coin selected is returned to the pocket before the next coin is selected. The Central Limit Theorem applies to both sampling with and without replacement, as long as the sample size is sufficiently large.
• Describe how sampling with replacement can affect the statistical properties of a sample, such as the sample variance.
  • Sampling with replacement allows for the possibility of the same value appearing multiple times in a sample. This can affect the statistical properties of the sample, such as the sample variance. When a value appears multiple times in a sample, it will have a greater influence on the sample variance compared to a sample where each value appears only once. This is because the sample variance is calculated as the average squared deviation from the sample mean, and a value that appears multiple times will contribute more to this calculation. As a result, sampling with replacement can lead to a sample variance that is different from the population variance, and this difference must be taken into account when making inferences about the population.
• Evaluate the role of sampling with replacement in simulation and bootstrapping techniques used to estimate the properties of a statistic.
  • Sampling with replacement plays a crucial role in simulation and bootstrapping techniques used to estimate the properties of a statistic, such as its standard error or confidence interval. In these techniques, random samples are drawn from the original data, with replacement, to create a large number of simulated or bootstrapped samples. The statistic of interest is then calculated for each of these samples, and the distribution of the statistic is used to estimate its properties. Sampling with replacement ensures that each simulated or bootstrapped sample has the same probability of being selected, which is necessary for these techniques to provide accurate estimates of the statistic's properties. Additionally, the ability to select the same value multiple times in the sampling process allows for the simulation of the variability that would be observed in repeated sampling from the population.

Related terms

Sampling Distribution: The probability distribution of a statistic, such as the sample mean or sample proportion, calculated from all possible samples of the same size that could be drawn from a population.

Central Limit Theorem: A fundamental result in probability and statistics that states that the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution, as the sample size increases.

Pocket Change: A practical example of sampling with replacement, where the amount of pocket change a person has can be considered a random variable that is sampled from the population of all possible amounts of pocket change.