Question

A small city contains 20,000 voters. Use the random number table to identify the voters to be included in a random sample of \(n=15\)

Short answer

Question: Explain the process of using a random number table to select a random sample of 15 voters from a small city with 20,000 voters.

Step-by-step solution

Understand the random number table

A random number table is a table filled with randomly generated numbers. In order to use the table to select our voters, we will assign an ID number to each voter in the city, then use the table to draw our sample.

Assign identification numbers to the voters

Since the city contains 20,000 voters, we will assign a unique 5-digit ID number to each voter. The ID numbers will range from 00001 to 20000. This way, every voter has an equal chance of being selected in our random sample.

Use the random number table to select voters

To select the random sample of 15 voters, follow these steps using the random number table: 1. Start at any row and column of the random number table. 2. Read 5-digit numbers from the table, ignoring any numbers that are not within the range of the assigned 5-digit ID numbers (00001 to 20000). 3. Once you find a valid 5-digit number from the table, identify the corresponding voter with that ID number and include them in your sample. 4. Continue moving through the table, selecting valid ID numbers until you have 15 unique voters in your sample.

Confirm your 15-member random sample

After using the random number table to select 15 unique voters, you should have a complete random sample. Ensure that each member of the sample has been assigned an identification number and that there are no duplicates. Your sample is now ready for further analysis or research.

Key concepts

Random Number Table

When conducting research, particularly in the field of probability and statistics, the use of a random number table is invaluable. In essence, it is a grid of digits, each randomly generated, which allows researchers to make selections in an unbiased manner. This tool ensures that every element in a population has an equal opportunity to be chosen for a sample.

The process of drawing samples with a random number table begins by matching each element in the population with a unique numerical identifier. When a population size is large, like the 20,000 voters in our example, a five-digit identification system is useful. Starting at any position in the table, five-digit numbers corresponding to our voter IDs are chosen sequentially while disregarding numbers outside our range. This ensures that our sample is random and representative of the entire population, thereby maintaining the integrity of statistical analyses.

Sampling Techniques

Sampling techniques are critical in research because they dictate how a subset of individuals, items, or events from a larger population are selected for analysis. These techniques can be broadly categorized into probability and non-probability sampling. Probability sampling gives each member of the population a known chance to be included in the sample, as is the case with random sampling. Other methods like stratified, cluster, and systematic sampling fall under this umbrella.

It's worth noting that probability sampling techniques, such as using a random number table, are known for their fairness and objectivity. They reduce selection bias and increase the likelihood that the sample will accurately reflect the population's properties. This in turn, enhances the reliability of the study's conclusions.

Probability and Statistics

The fields of probability and statistics are intimately connected, with probability theory underpinning statistical practice. Probability deals with the likelihood of events occurring based on known parameters, while statistics uses this framework to analyze and interpret data collected from samples to make inferences about entire populations.

Random sampling, the cornerstone of statistical methodology, leverages probability to ensure all possible samples of a given size have an equal chance of being picked. This is crucial when enabling researchers to generalize their findings from the sample to the larger population with a quantifiable degree of certainty. Therefore, the use of probability theory through tools such as random number tables is a mainstay in statistics, strengthening the validity of the research conducted.

Random Sample Selection

Selecting a random sample is a meticulous process that determines the quality of the data collected. The random sample selection method ensures that the sample is an unbiased representation of the larger group. This involves assigning every member an equal probability of being selected, then using mechanisms like a random number table or computerized random number generators to make the actual selection.

For example, in our scenario with 20,000 voters, assigning a unique identifier to each voter allows for an orderly selection process. Such a method preserves the randomness and avoids the introduction of systematic error. It's also crucial to verify that each selected member is unique, preventing any individual from being included more than once. This is key to upholding the integrity of the sampling and consequent research findings.