Consider a population of size four, the members of which...

Consider a population of size four, the members of which have values $x_{1}, x_{2}, x_{3}, x_{4}$ a. If simple random sampling were used, how many samples of size two are there? b. Suppose that rather than simple random sampling, the following sampling scheme is used. The possible samples of size two are $$\left\{x_{1}, x_{2}\right\},\left\{x_{2}, x_{3}\right\},\left\{x_{3}, x_{4}\right\},\left\{x_{1}, x_{4}\right\}$$ and the sampling is done in such a way that each of these four possible samples is equally likely. Is the sample mean unbiased?

Consider a population of size four, the members of which have values $x_{1}, x_{2}, x_{3}, x_{4}$
a. If simple random sampling were used, how many samples of size two are there?
b. Suppose that rather than simple random sampling, the following sampling scheme is used. The possible samples of size two are $$\left\{x_{1}, x_{2}\right\},\left\{x_{2}, x_{3}\right\},\left\{x_{3}, x_{4}\right\},\left\{x_{1}, x_{4}\right\}$$ and the sampling is done in such a way that each of these four possible samples is equally likely. Is the sample mean unbiased?

Key Concepts

Simple Random Sampling

Simple random sampling is a fundamental sampling technique where each possible sample of a given size is equally likely to be chosen from the population. This concept is crucial because it helps ensure that the sample is representative of the population, minimizing bias in the estimation of population parameters.

Combinatorial Calculations

Combinatorial calculations involve determining the number of ways to select items from a set, typically using binomial coefficients. In the context of sampling, it is used to calculate the total number of possible samples, which is important for assessing the likelihood of selecting a particular sample under a random sampling scheme.

Unbiased Estimator

An unbiased estimator is a statistical estimator whose expected value is equal to the true value of the parameter it is estimating. This means that on average, the estimator does not overestimate or underestimate the parameter, making it desirable for inferential statistics in both theory and application.

Sampling Design and Its Effects

The sampling design refers to how a sample is selected from the population, and it can influence the properties of the estimator. Even if a sampling scheme is not a simple random sample, the sample mean can still be unbiased provided that the expected value of the weighted contributions from all potential samples equals the true population mean. This emphasizes the importance of understanding the relationship between the sampling design and estimator properties such as bias.

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