Question

A survey is being conducted in a city with 1 million residents. It would be far too

expensive to survey all of the residents, so a random sample of size 1000 is chosen
(in practice, there are many challenges with sampling, such as obtaining a complete
list of everyone in the city, and dealing with people who refuse to participate). The
survey is conducted by choosing people one at a time, with replacement and with equal
probabilities. Explain how sampling with vs. without replacement here relates to the birthday problem.

Answer 1

Sampling with and without replacement in a survey relates to the birthday problem through the concept of independent events. In sampling with replacement, each member of the population has the possibility of being chosen more than once, and the events are considered to be independent.

Step-by-step explanation:

Sampling with and without replacement in a survey relates to the birthday problem through the concept of independent events. In sampling with replacement, each member of the population has the possibility of being chosen more than once, and the events are considered to be independent.

This is similar to the birthday problem, where the probability of two people having the same birthday is independent for each pair of people. On the other hand, in sampling without replacement, each member of the population can be chosen only once, and the events are considered not to be independent.

This is similar to the birthday problem when we consider a group of people where each person can only have one birthday.The question relates to the concept of sampling and how different methods of sampling can affect the outcomes of a survey or a study.

In sampling with replacement, after a person is chosen for the survey, they are 'replaced' back into the population, making it possible for them to be selected again. This mirrors the birthday problem, where one is interested in the probability that, in a set of randomly chosen people, some pair of them will have the same birthday.

With sampling without replacement, once an individual is selected for the survey, they cannot be selected again, making each selection dependent on the previous selections and reducing the probability of picking the same person again, unlike in the birthday problem which assumes independence between individuals' birthdays.