Sampling Without Replacement

Definition

Sampling without replacement is a method of drawing samples from a population where each item is removed from the population after it is selected, ensuring that no item can be selected more than once. This is in contrast to sampling with replacement, where items are returned to the population after being selected, allowing for the possibility of the same item being chosen multiple times.

5 Must Know Facts For Your Next Test

1. Sampling without replacement is commonly used in situations where the population size is finite and the goal is to obtain a representative sample without duplicates.
2. In sampling without replacement, the probability of selecting an item decreases with each successive draw, as the number of available items in the population decreases.
3. Sampling without replacement is often used in hypothesis testing, where the goal is to make inferences about a population based on a sample that accurately represents the population.
4. The hypergeometric distribution is a probability distribution that models the number of successes in a fixed number of draws without replacement from a finite population.
5. Sampling without replacement is important in the context of independent and mutually exclusive events, as it ensures that the probability of each event is independent of the others.

Review Questions

• Explain how sampling without replacement differs from sampling with replacement and how this impacts the probability of selecting an item.
  • In sampling with replacement, each item in the population has an equal probability of being selected, and the same item can be chosen multiple times. In contrast, sampling without replacement removes each selected item from the population, so the probability of selecting an item decreases with each successive draw as the available items in the population decrease. This means that the probability of selecting an item in sampling without replacement is dependent on the previous selections, unlike in sampling with replacement where the probabilities are independent.
• Describe the relationship between sampling without replacement and the hypergeometric distribution, and explain how this distribution can be used to model the probability of successes in a fixed number of draws.
  • The hypergeometric distribution is a probability distribution that models the number of successes in a fixed number of draws without replacement from a finite population. In the context of sampling without replacement, the hypergeometric distribution can be used to calculate the probability of obtaining a certain number of successes (e.g., the number of items with a particular characteristic) in a sample drawn from a population, given the total population size, the number of items with the characteristic, and the sample size. This distribution is particularly useful when the population size is finite and the goal is to make inferences about the population based on a representative sample.
• Analyze how sampling without replacement is important in the context of independent and mutually exclusive events, and explain how it ensures the independence of the probabilities of these events.
  • Sampling without replacement is crucial in the context of independent and mutually exclusive events because it ensures that the probability of each event is independent of the others. When sampling without replacement, the probability of selecting an item decreases with each successive draw, and the selected items are removed from the population. This means that the probability of an event occurring is not affected by the occurrence of previous events, as the pool of available items is continuously changing. This independence of events is a fundamental assumption in the study of probability and statistical inference, and sampling without replacement helps maintain this independence, leading to more accurate analysis and conclusions.

Related terms

Sampling with Replacement: A sampling method where items are returned to the population after being selected, allowing for the possibility of the same item being chosen multiple times.

Hypergeometric Distribution: A discrete probability distribution that describes the probability of a certain number of successes in a fixed number of draws, without replacement, from a finite population.

Independent Events: Events where the occurrence of one event does not affect the probability of the other event occurring.