Simple Random Sampling and Random Number Table Application Tables

Concepts of Simple Random Sampling

Simple Random Sample (SRS): A simple random sample of size $n$ from a population of size $N$ is a sample selected in such a way that every possible sample of size $n$ has an equally likely chance of being chosen.
Equality of Opportunity: In an SRS, every individual in the population has the same probability of being included in the sample.
Sampling Frame: To select a simple random sample, a list of all individuals in the population is required. This list is known as the sampling frame.
Selection Methods: Simple random samples are typically obtained using:
- Random Number Tables: Printed tables of digits (0-9) generated in a way that each digit is equally likely to appear in any position.
- Random Number Generators: Software or calculators that use algorithms to produce a sequence of numbers that lack any pattern.

Problem Overview: Faculty Opinion Survey

Population ( $N$ ): The community college employs $N = 86$ full-time faculty members.
Sample Size ( $n$ ): The college president wants to select a sample size of $n = 9$ faculty members.
Assignment of Numbers: The faculty members are numbered from $1$ to $86$ .
- Note on Numbering: Since the highest number is $86$ (a two-digit number), all members should be assigned two-digit labels from $01$ to $86$ for consistency when using a random number table.

Methodology: Using a Random Number Table

Objective: Identify $9$ unique numbers between $01$ and $86$ using the provided table.
Starting Point: Row 3, Column 6.
Direction of Selection: Proceed downward through the columns.
Rules for Selection:
- Two-Digit Selection: Because the population size $N = 86$ consists of two digits, the digits from the table must be read in pairs (using columns 6 and 7).
- In-Range Criteria: Only numbers between $01$ and $86$ inclusive are selected.
- Out-of-Range Criteria: If a number is greater than $86$ (e.g., $87, 88, 99$ ) or is $00$ , it must be discarded.
- Uniqueness Criteria: If a number that has already been chosen appears again, it is ignored (sampling without replacement).

Execution of the Sampling Process

Step-by-Step Selection

Based on the provided "Random Numbers" icon view, we locate the digits starting at Row 3 in the group for Columns 06-10.

Random Number Table Snippet (Columns 06-10):
- Row 01: 16033 (First digits are 16)
- Row 02: 64413 (First digits are 64)
- Row 03: 87817 (First digits are 87)
- Row 04: 64089 (First digits are 64)
- Row 05: 07948 (First digits are 07)
- Row 06: 93283 (First digits are 93)

Applying the Criteria

Row 3, Columns 06-07: The digits are 87.
- Action: Discard, because 87 > 86.
Row 4, Columns 06-07: The digits are 64.
- Action: Keep (1st member). The value $64$ is within the range $01-86$ .
Row 5, Columns 06-07: The digits are 07.
- Action: Keep (2nd member). The value $07$ is within the range $01-86$ .
Row 6, Columns 06-07: The digits are 93.
- Action: Discard, because 93 > 86.

Note: To complete the full sample of 9, the surveyor would continue to the next available block of columns or the next page of the table following the same downward pattern and selection rules until 9 unique faculty numbers are obtained.

Implications and Ethical Considerations

Bias Mitigation: Using a random number table eliminates researcher bias in selecting participants. If the president chose faculty members personally, they might subconsciously select those who agree with the building project.
Representativeness: While an SRS is mathematically fair, there is still a small chance of "sampling error," where the random sample does not perfectly reflect the demographics of the whole faculty (e.g., all selected members might be from the same department by pure chance).
Anonymity and Confidentiality: In surveys regarding institutional projects, ensuring the random selection is coupled with confidential data collection is vital for obtaining honest opinions from faculty.

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