Sampling Without Replacement

Definition

Sampling without replacement is a statistical technique where items or individuals are selected from a finite population, and once an item is selected, it is not returned to the population before the next selection. This method ensures that each item in the population has a unique chance of being chosen and prevents the same item from being selected multiple times within a single sample.

5 Must Know Facts For Your Next Test

1. Sampling without replacement is commonly used when the population size is relatively small, and the researcher wants to ensure that each item or individual has an equal chance of being selected.
2. The hypergeometric distribution is used to model the probability of success in sampling without replacement, where the number of successes in the sample is the random variable.
3. The finite population correction factor is used to adjust the standard error of a sample statistic when the sample is drawn from a finite population, to account for the fact that the population size is limited.
4. Sampling without replacement is often preferred over sampling with replacement when the population size is small, as it helps to ensure that the sample is representative of the entire population.
5. The use of sampling without replacement can have implications for the statistical analysis and inferences made from the sample data, as it affects the underlying probability distributions and the standard errors of the sample statistics.

Review Questions

• Explain how sampling without replacement differs from sampling with replacement and how it impacts the probability distributions used in statistical analysis.
  • In sampling without replacement, once an item or individual is selected from the population, it is not returned before the next selection, ensuring that each item has a unique chance of being chosen. This contrasts with sampling with replacement, where the selected item is returned to the population before the next selection, allowing the same item to be chosen multiple times. The key difference is that sampling without replacement follows the hypergeometric distribution, while sampling with replacement follows the binomial distribution. This distinction is important because it affects the underlying probability calculations and the standard errors of the sample statistics.
• Describe the role of the finite population correction factor in the context of sampling without replacement and explain how it impacts the statistical inferences made from the sample data.
  • The finite population correction factor is used to adjust the standard error of a sample statistic when the sample is drawn from a finite population, as is the case in sampling without replacement. This adjustment is necessary because the population size is limited, and the sample is not drawn from an infinitely large population. The finite population correction factor reduces the standard error of the sample statistic, which in turn affects the confidence intervals and hypothesis testing procedures used to make statistical inferences. Ignoring the finite population correction factor can lead to overly conservative conclusions, as it fails to account for the reduced variability in the sample due to the limited population size.
• Analyze the advantages and disadvantages of using sampling without replacement compared to sampling with replacement, and discuss the scenarios where each method may be more appropriate.
  • The primary advantage of sampling without replacement is that it ensures each item or individual in the population has a unique chance of being selected, which can be important when the population size is relatively small. This method helps to ensure the sample is representative of the entire population. However, the disadvantage is that it can be more computationally intensive and may require more complex statistical analysis, as the underlying probability distribution (hypergeometric) is more complex than the binomial distribution used in sampling with replacement. Sampling without replacement is generally more appropriate when the population size is small, and the researcher wants to ensure each item has an equal chance of being selected. Sampling with replacement, on the other hand, may be more appropriate when the population size is large, and the researcher is primarily interested in the overall statistical properties of the sample rather than the specific items or individuals selected.

Related terms

Sampling with Replacement: A sampling method where an item or individual is selected from a population, recorded, and then returned to the population before the next selection. This allows the same item to be selected multiple times within a single sample.

Hypergeometric Distribution:

A discrete probability distribution that describes the probability of a certain number of successes in a fixed number of trials without replacement from a finite population.

Finite Population Correction Factor:

A statistical adjustment made to the standard error of a sample statistic when the sample is drawn from a finite population, to account for the fact that the population size is limited.