Hypergeometric Distribution

Definition

The hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. It is commonly used to model situations where a sample is drawn from a population without replacement, such as selecting balls from an urn or cards from a deck.

5 Must Know Facts For Your Next Test

1. The hypergeometric distribution is used to model situations where a sample is drawn from a finite population without replacement, such as selecting balls from an urn or cards from a deck.
2. The key parameters of the hypergeometric distribution are the population size (N), the number of successes in the population (K), and the sample size (n).
3. The hypergeometric distribution is a discrete probability distribution, meaning it can only take on integer values, unlike the continuous normal distribution.
4. The hypergeometric distribution is often used in quality control, sampling, and hypothesis testing applications where the population size is finite and the sampling is done without replacement.
5. The hypergeometric distribution is related to the binomial distribution, but the key difference is that the binomial distribution assumes independent trials with replacement, while the hypergeometric distribution assumes sampling without replacement from a finite population.

Review Questions

• Explain how the hypergeometric distribution differs from the binomial distribution and the key parameters that define it.
  • The key difference between the hypergeometric distribution and the binomial distribution is that the hypergeometric distribution models sampling without replacement from a finite population, while the binomial distribution assumes independent trials with replacement. The parameters that define the hypergeometric distribution are the population size (N), the number of successes in the population (K), and the sample size (n). These parameters determine the probability of observing a certain number of successes in the sample drawn from the finite population.
• Describe a real-world scenario where the hypergeometric distribution would be an appropriate model and explain how the parameters of the distribution would be determined.
  • A real-world scenario where the hypergeometric distribution would be an appropriate model is quality control testing of a batch of manufactured products. Suppose a manufacturer has produced a batch of 1,000 items (population size, N = 1,000), and they know that 50 of these items are defective (number of successes in the population, K = 50). The manufacturer wants to randomly select a sample of 20 items (sample size, n = 20) to test for defects. The hypergeometric distribution would be the appropriate model to determine the probability of observing a certain number of defective items in the sample, as the sampling is done without replacement from the finite population of manufactured items.
• Analyze how the hypergeometric distribution can be used in hypothesis testing to determine the likelihood of observing a certain number of successes in a sample drawn from a finite population without replacement.
  • The hypergeometric distribution can be used in hypothesis testing to determine the likelihood of observing a certain number of successes in a sample drawn from a finite population without replacement. For example, a researcher may want to test the hypothesis that a certain proportion of a population possesses a particular characteristic. By modeling the number of successes (individuals with the characteristic) in the sample using the hypergeometric distribution, the researcher can calculate the probability of observing the given number of successes under the null hypothesis. This probability can then be used to determine whether the observed sample results are statistically significant and to make inferences about the population proportion.

Related terms

Binomial Distribution: A discrete probability distribution that describes the number of successes in a sequence of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure).

Sampling Without Replacement: A sampling technique where the selected items are not returned to the population before the next item is selected, resulting in a change in the probability of selection for the remaining items.

Finite Population: A population with a known, fixed number of elements, in contrast to an infinite population where the number of elements is unknown or unbounded.