Simple Random Sampling Uses A Sample Of Size N From A Popula

Simple Random Sampling Uses A Sample Of Size N From a Population Of

Simple random sampling involves selecting a subset of individuals or items from a larger population in such a way that every possible sample of a given size has an equal chance of being chosen. This method is widely used in statistics to make inferences about population characteristics based on sample data. When conducting a simple random sample, the goal is to ensure that the sample accurately reflects the population, minimizing bias and enabling valid conclusions.

In the context of sampling from a population of 45 bank accounts, where a sample of 6 accounts is drawn to study the populations' characteristics, a natural question arises: How many different samples of a certain size can be formed? Specifically, if we are interested in the number of samples of four accounts that can be selected from these 45 accounts, this problem can be addressed using combinatorial methods, particularly the binomial coefficient or "n choose k" formula.

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The problem of determining the number of possible samples is rooted in combinatorial mathematics, which provides formulas for counting the number of ways to select items from a larger set without regard to the order of selection. The binomial coefficient, often read as "n choose k," where n is the total number of items in the population and k is the size of the sample, is given by the formula:

C(n, k) = n! / (k! * (n - k)!)

where "!" denotes factorial, the product of all positive integers up to that number. This formula calculates the total number of unique combinations of k items that can be selected from a set of n items, ensuring each combination is counted only once, regardless of order.

Applying this to the problem at hand, with N = 45 and n = 6, the question asks: How many samples of 4 accounts can be formed? Using the binomial coefficient, the calculation becomes:

C(45, 4) = 45! / (4!  (45 - 4)!) = 45! / (4!  41!)

Calculating the factorial values is straightforward but can be simplified using software or calculator functions designed for combinatorics. The result provides the total number of ways to select 4 accounts from the population of 45.

This combinatorial approach highlights the importance of understanding the underlying probabilities and counting methods in statistical sampling. Accurate enumeration of possible samples allows researchers to assess variability, calculate probabilities, and make inferences about the population based on sample data.

In practical applications, such as quality control, opinion surveys, or financial analyses, the principles of simple random sampling combined with combinatorial mathematics underpin the design of experiments and the interpretation of their results. Recognizing the total number of possible samples enables statisticians to evaluate the likelihood of observing particular sample characteristics and to estimate population parameters with confidence.

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