Simple Random Sampling With Replacement Example

Simple Random Sampling With Replacement: A Comprehensive Guide with Examples

Simple random sampling with replacement (SRSWR) is a fundamental probability sampling technique used extensively in statistics and research. Understanding its mechanics, applications, and limitations is crucial for anyone involved in data analysis or research design. This comprehensive guide will delve into the intricacies of SRSWR, providing clear explanations, illustrative examples, and insights into its practical applications.

What is Simple Random Sampling With Replacement (SRSWR)?

Simple random sampling with replacement (SRSWR) is a sampling method where each member of the population has an equal chance of being selected, and each selection is independent of the others. The crucial aspect of "with replacement" means that once a member is selected, it's returned to the population pool before the next selection. This allows for the possibility of selecting the same member multiple times.

This contrasts with simple random sampling without replacement (SRSWOR), where a selected member is removed from the population pool, preventing its re-selection. While both methods are random, the "with replacement" aspect significantly impacts the statistical properties of the sample and the subsequent analysis.

Key Characteristics of SRSWR:

• Equal Probability: Every member of the population has an identical probability of being chosen in each draw.
• Independence: The selection of one member doesn't influence the probability of selecting any other member. Each draw is an independent event.
• Replacement: Selected members are returned to the population before the subsequent draw, allowing for duplicates in the sample.

Why Use SRSWR?

While SRSWOR is often preferred in many situations, SRSWR offers specific advantages:

• Simplicity: It's conceptually simpler to understand and implement than other sampling techniques. The process is straightforward and easy to explain.
• Theoretical Convenience: SRSWR simplifies many statistical calculations and theoretical derivations. It often leads to simpler formulas and easier interpretations of results. This is particularly useful in probability theory and statistical modeling.
• Suitable for Large Populations: When dealing with extremely large populations, the difference between sampling with and without replacement becomes negligible. The probability of selecting the same member twice becomes incredibly small, making SRSWR a practical choice.
• Estimation of Proportions: SRSWR provides unbiased estimates of population proportions, even with a relatively small sample size.

Illustrative Examples of SRSWR

Let's explore several scenarios to illustrate the concept and applications of SRSWR.

Example 1: Drawing Marbles from a Bag

Imagine a bag containing 5 marbles: 2 red and 3 blue. We want to select a sample of 2 marbles using SRSWR.

Process:

1. We draw one marble. The probability of drawing a red marble is 2/5, and the probability of drawing a blue marble is 3/5.
2. We record the color of the marble.
3. We return the marble to the bag.
4. We draw a second marble. The probabilities remain the same as in step 1, regardless of the color of the first marble drawn.

Possible Outcomes:

• RR: Red, Red (Probability: (2/5) * (2/5) = 4/25)
• RB: Red, Blue (Probability: (2/5) * (3/5) = 6/25)
• BR: Blue, Red (Probability: (3/5) * (2/5) = 6/25)
• BB: Blue, Blue (Probability: (3/5) * (3/5) = 9/25)

Notice that the probabilities sum to 1 (4/25 + 6/25 + 6/25 + 9/25 = 25/25 = 1). The independence of each draw is evident.

Example 2: Customer Satisfaction Survey

A company wants to assess customer satisfaction. They have a database of 10,000 customers. They decide to select a sample of 100 customers using SRSWR to conduct a satisfaction survey.

Process:

1. Each customer is assigned a unique number from 1 to 10,000.
2. A random number generator is used to select 100 numbers between 1 and 10,000.
3. The customers corresponding to the selected numbers are surveyed. If the same number is generated twice, the same customer is surveyed again.

This approach ensures that each customer has an equal chance of being selected, and the selection of one customer does not influence the selection of another.

Example 3: A/B Testing

In A/B testing, two versions of a website or app (A and B) are presented to users. SRSWR can be used to assign users to either version A or version B.

Imagine 1000 users. Using SRSWR, each user has a 50% chance of being assigned to version A and a 50% chance of being assigned to version B. This allows for the possibility of the same user being assigned to both versions (though this is unlikely with a large user base).

Calculating Probabilities in SRSWR

Calculating probabilities in SRSWR is straightforward due to the independence of each draw. The probability of a specific sequence of selections is simply the product of the individual probabilities of each selection.

For example, if we have a population of size N and we want to select a sample of size n with replacement, the probability of obtaining a particular sample is:

(1/N) * (1/N) * ... * (1/N) = (1/N)^n

where (1/N) represents the probability of selecting any single member of the population in a single draw.

Advantages and Disadvantages of SRSWR

Advantages:

• Simplicity and Ease of Implementation: The procedure is easily understood and executed.
• Unbiased Estimates: Provides unbiased estimates of population parameters, particularly proportions.
• Theoretical Tractability: Simplifies statistical calculations and theoretical analysis.

Disadvantages:

• Possible Duplicates: The possibility of selecting the same member multiple times can lead to less efficient use of sampling resources.
• Higher Variance: Compared to SRSWOR, SRSWR generally has higher variance in the sample statistics, making it less precise for estimation. This means that the sample mean, for example, may be further from the true population mean.
• Not Ideal for Small Populations: The probability of duplicates significantly increases with small populations, making SRSWOR generally preferable in these cases.

When to Use SRSWR

SRSWR is most appropriate in the following situations:

• Large Populations: When the population size is very large, the probability of selecting the same member twice becomes negligible.
• Theoretical Studies: In theoretical statistical studies, SRSWR often simplifies mathematical derivations and provides more manageable models.
• Situations where Duplicates are Acceptable: In some scenarios, selecting the same member multiple times doesn't pose a problem. For instance, in A/B testing, it might not be critical if a user is assigned to the same version twice.
• Estimating Proportions: SRSWR provides unbiased estimates of population proportions, making it a suitable choice for surveys and polls.

Comparison with SRSWOR

The following table summarizes the key differences between SRSWR and SRSWOR:

Conclusion

Simple random sampling with replacement is a valuable technique in probability sampling, especially when dealing with large populations or when theoretical simplicity is prioritized. Understanding its characteristics, advantages, disadvantages, and appropriate applications allows researchers and data analysts to make informed decisions about their sampling strategies and ensure the validity and reliability of their results. By carefully considering the specific context and objectives of a study, one can choose between SRSWR and other sampling methods to obtain the most effective and efficient results. Remember to always consider the potential implications of duplicates when choosing between sampling with and without replacement.