Understanding Simple Random Sampling with Replacement

Simple random sampling is a fundamental technique in statistics used to select a sample from a population. When sampling is done with replacement, it means that once an item is selected and included in the sample, it is put back into the population and can be selected again. This means the same item can appear multiple times in a single sample.

The question asks for the total number of distinct possible samples of a specific size, say 'n', that can be drawn from a population of size 'N' when using simple random sampling with replacement. In this type of sampling, the order in which the items are selected matters. For example, selecting item A then item B is considered a different sample than selecting item B then item A, even if the items are put back each time.

Calculating the Number of Samples with Replacement

Let's think about the selection process step-by-step for simple random sampling with replacement:

• For the first selection, there are N possible items that can be chosen from the population.
• Since the item is replaced, for the second selection, there are still N possible items that can be chosen.
• This process continues for each of the 'n' selections.

Since there are N choices for each of the 'n' selections, and these selections are independent (because of replacement), the total number of possible sequences of selections (samples where order matters and replacement is allowed) is the product of the number of choices at each step.

Total number of samples = (Number of choices for 1st selection) \(\times\) (Number of choices for 2nd selection) \(\times \dots \times\) (Number of choices for nth selection)

Total number of samples = N \(\times\) N \(\times \dots \times\) N (n times)

This product can be written in a more compact form using exponents:

Total number of samples = Nⁿ

Analyzing the Options

Let's examine the given options in the context of simple random sampling with replacement:

1. ^NP_n (Permutations of N items taken n at a time): This formula, given by \(\text{N! / (N-n)!}\), calculates the number of ways to arrange 'n' items selected from 'N' items without replacement where order matters. This is not applicable when replacement is allowed.
2. ^NC_n (Combinations of N items taken n at a time): This formula, given by \(\text{N! / (n! * (N-n)!)} \), calculates the number of ways to select 'n' items from 'N' items without replacement where order does not matter. This is not applicable when replacement is allowed and order matters (as it does in the sequence of selections in sampling).
3. Nⁿ: As derived above, this formula represents the number of ways to select 'n' items from 'N' possibilities with replacement, where the order of selection matters. This perfectly matches the description of simple random sampling with replacement.
4. None of the options: Since Nⁿ is the correct formula, this option is incorrect.

Therefore, the number of samples of size n when performing simple random sampling with replacement from a population of size N is Nⁿ.

Example for Simple Random Sampling with Replacement

Consider a population of size N=3, with elements {A, B, C}. We want to find the number of samples of size n=2 using simple random sampling with replacement.

Using the formula Nⁿ, the number of samples should be 3² = 9.

Let's list the possible samples:

• (A, A)
• (A, B)
• (A, C)
• (B, A)
• (B, B)
• (B, C)
• (C, A)
• (C, B)
• (C, C)

There are indeed 9 distinct ordered samples when sampling with replacement, confirming the formula Nⁿ.

Revision Table: Sampling Methods

Additional Information: Concepts in Sampling

• Population: The entire group of individuals or items that you want to study.
• Sample: A subset of the population from which you collect data.
• Sampling Unit: An individual member of the population.
• Sampling Frame: A list of all the sampling units in the population from which the sample is drawn.
• Simple Random Sampling: A method where every possible sample of the desired size has an equal chance of being selected.
• Sampling with Replacement: A method where a selected unit is returned to the population before the next selection, allowing it to be selected again.
• Sampling without Replacement: A method where a selected unit is not returned to the population, meaning it cannot be selected again.
• Understanding whether sampling is with or without replacement is crucial as it affects the probability calculations and the formula for the number of possible samples.