Question

In Simple Random Sampling with replacement (SRSWR), the total no. of samples (with standard notation) will be
1. \(\binom{N}{n}\)
2. \(N^n\)
3. n + N
4. n*N

• A. \(\binom{N}{n}\)
• B. \(N^n\)
• C. n + N
• D. n*N

Hint:

Remember:
- With Replacement (SRSWR) \(\rightarrow\) Order matters, units can repeat \(\rightarrow N^n\).
- Without Replacement (SRSWOR) \(\rightarrow\) Order does not matter, units cannot repeat \(\rightarrow \binom{N}{n}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question relates to sampling theory, specifically comparing the combinations of possible samples under Simple Random Sampling with Replacement (SRSWR) versus Simple Random Sampling without Replacement (SRSWOR).

Step 2: Detailed Explanation:
Let us define the parameters and analyze the sampling procedures:
Let \( N \) be the total number of units in the population, and let \( n \) be the size of the sample to be drawn.
- Simple Random Sampling with Replacement (SRSWR):
In this method, a unit is selected at random from the population, its characteristics are recorded, and it is returned to the population before the next unit is drawn.
This means that at each of the \( n \) draws, every one of the \( N \) population units has an equal probability of being selected.
- For the \( 1^{\text{st}} \) draw, there are \( N \) choices.
- For the \( 2^{\text{nd}} \) draw, there are \( N \) choices.
- For the \( n^{\text{th}} \) draw, there are \( N \) choices.
By the fundamental counting principle, the total number of distinct ordered samples of size \( n \) is:
\[ N \times N \times \dots \times N \text{ (n times)} = N^n \] - Simple Random Sampling without Replacement (SRSWOR):
If the sampling were without replacement, once a unit is selected, it is not returned. The total number of unique samples of size \( n \) would be:
\[ \binom{N}{n} \] Since the question specifies Simple Random Sampling with replacement (SRSWR), the total number of samples is \( N^n \).

Step 3: Final Answer:
The total number of samples in SRSWR is \( N^n \).
Therefore, the correct choice is Option (B).