Calculating Probability in SRSWOR

Let's break down how to find the probability of selecting a specific sample when using Simple Random Sampling Without Replacement (SRSWOR).

In this problem, we are given:

• Population size (\(N\)): The total number of units in the population is 10.
• Sample size (\(n\)): We are selecting a sample of 2 units.
• Sampling Method: SRSWOR (Simple Random Sampling Without Replacement). This means that once a unit is selected for the sample, it is not put back into the population, so it cannot be selected again. Crucially, in SRSWOR, every possible sample of size \(n\) from the population of size \(N\) has an equal chance of being selected.

Finding the Total Number of Possible Samples

To find the probability of selecting a specific sample, we first need to determine the total number of distinct samples of size \(n\) that can be drawn from a population of size \(N\) using SRSWOR. The number of ways to choose \(n\) items from \(N\) distinct items without regard to the order of selection and without replacement is given by the combination formula:

\[ \binom{N}{n} = \frac{N!}{n!(N-n)!} \]

In our case, \(N = 10\) and \(n = 2\). So, the total number of possible samples of size 2 from a population of 10 is:

\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2!8!} \]

Let's calculate this value:

\[ \frac{10 \times 9 \times 8!}{2 \times 1 \times 8!} = \frac{10 \times 9}{2} = \frac{90}{2} = 45 \]

So, there are 45 different possible samples of 2 units that can be drawn from the population of 10 units using SRSWOR.

Calculating the Probability of a Specified Sample

In SRSWOR, each of the possible samples has an equal probability of being selected. Since there are 45 possible samples, the probability of selecting any one specific sample is 1 divided by the total number of possible samples.

Probability of selecting a specified sample = \( \frac{1}{\text{Total number of possible samples}} \)

Probability = \( \frac{1}{45} \)

Therefore, the probability of selecting a specified sample of 2 units from a population of 10 units using SRSWOR is \(1/45\).

Revision Table: SRSWOR Key Points

Additional Information: Why SRSWOR is Important

Simple Random Sampling Without Replacement (SRSWOR) is a fundamental sampling method in statistics and survey methodology. Here's why it's important:

• Fairness: It ensures that every unit in the population has an equal chance of being included in the sample at the initial draw, and that every possible combination of units (sample) has an equal chance of being selected.
• Unbiased Estimates: When done correctly, SRSWOR leads to unbiased estimators for population parameters like the population mean or total.
• Basis for Inference: It provides a strong theoretical foundation for statistical inference, allowing us to calculate standard errors and construct confidence intervals.
• Efficiency: Compared to sampling with replacement, SRSWOR is often more efficient, especially when the sample size is a significant fraction of the population size, as it prevents the same unit from being sampled multiple times within the same sample.

Understanding the probability associated with selecting specific samples is crucial for understanding the properties of estimators derived from SRSWOR samples.