Sampling with Replacement

In sampling with replacement, the units drawn are returned to the population before drawing the next unit. This means the same individual can be chosen more than once in the sampling process. The sampling with replacement may provide valuable insights while maintaining flexibility in selecting samples from a given population.

Key Characteristics of Sampling with Replacement

The following are key characteristics of Sampling with Replacement:

1. Independence: Each selection is independent, as the same item can be selected multiple times.
2. Population Size: The effective population size remains the same for each draw since previously selected items are replaced.
3. Use Cases: This method is commonly used in algorithms, simulations, and bootstrapping techniques in statistics, where it’s important to assess variability or make inferences from a sample.

Example of Sampling with Replacement

As an example of sampling with replacement, suppose, you have a bag containing three colored balls (red, blue, and green), and you sample with a replacement, if you draw a red ball, you put it back into the bag before the next draw. As a result, in subsequent draws, you could again draw a red ball.

Drawing All Possible Samples Using Sampling with Replacement

Question: Consider a population with elements A, B, C, and D. Draw all possible samples of size 2 with replacement from this population.

Solution: In this problem, $N=4$ and $n=2$.

Possible number of samples (with replacement) = $N^n = 4^2 = 16$.

The 16 samples of size 2 are

AA	AB	AC	AD
BA	BB	BC	BD
CA	CB	CC	CD
DA	DB	DC	DD

Question: Draw all possible samples of size 3 with replacement from a population having elements 2, 4, and 6.

Solution:

Population size = $N=3$, Sample size = n = 3$

Number of possible samples are $N^n = 3^3 = 27$

There are two ways to list these samples.

First Method:

First divide possible samples (27) by the population size unit quotient 1 is returned. For example, $\frac{27}{3} = 9, \quad \frac{9}{3}, \quad \frac{9}{3}=1$.

We obtained three quotients: 9, 3, and 1. These are the number of repetitions of population units. First, write every unit 9 times, then 3 times, and lastly, write every unit 1 time.

Second Method:

First, make the samples of size 2, which are easy to draw.

2, 2
2, 4
2, 6
4, 2
4, 4
4, 6
6, 2
6, 4
6, 6

Repeat these samples three times. Since the required number of samples is 27, add every population unit at (the start or) at the end of these samples of size two.

2, 2, 2	2, 2, 4	2, 2, 6
2, 4, 2	2, 4, 4	2, 4, 6
2, 6, 2	2, 6, 4	2, 6, 6
4, 2, 2	4, 2, 4	4, 2, 6
4, 4, 2	4, 4, 4	4, 4, 6
4, 6, 2	4, 6, 4	4, 6, 6
6, 2, 2	6, 2, 4	6, 2, 6
6, 4, 2	6, 4, 4	6, 4, 6
6, 6, 2	6, 6, 4	6, 6, 6

From the table above, 2 is added in the last of the first nine samples, then 4 is added in the last of the next 9 samples and finally 6 is added in the last nine samples.

Real-Life Examples of Sampling with Replacement

The following are some real-life examples of sampling with replacement:

1. Lottery Draws: In some types of lotteries, numbers can be drawn multiple times before the final selection. For example, if a lottery allows for the same number to be drawn again after being selected, this is akin to sampling with replacement.
2. Quality Control in Manufacturing: In a factory, inspectors might draw samples of products to test for defects. After testing each item, they return it to the production line before drawing the next sample to maintain the same population size and ensure each product has a chance of being selected again.
3. Genetic Studies: In genetics, researchers might take DNA samples from a population to study traits or disorders. By replacing each sample with the population (considering genetic diversity), they can analyze the data while allowing for the possibility of selecting the same individual multiple times.
4. Surveys: When conducting surveys, researchers might randomly select participants from a population (like voters or consumers) and, after querying each individual, they can include them again in the pool for subsequent selections, especially in larger datasets where the same individuals might provide valuable insights if repeated.
5. Educational Testing: In standardized testing, students might take multiple attempts at a test where scores from previous attempts can be considered again in analyses to assess trends in learning or improvement.
6. Customer Behavior Analysis: Companies may analyze customer purchase patterns by repeatedly sampling transactions. For instance, if a customer makes multiple purchases, their transaction data might be included in each analysis to understand their buying behavior over time.

Sampling Quiz Questions

Simulation and Sampling in R