Question

Consider a population consisting of 600 individuals with unique IDs: \(001,002, \ldots, 600 .\) Use the following string of random digits to select a simple random sample of 5 individuals. List the IDs of the individuals selected for your sample. $$ 728121876442121593787803547216596851 $$

Short answer

IDs of the individuals selected for the sample.

Step-by-step solution

Understanding the exercise

We are instructed to select a sample of 5 individuals from a population of 600 using the given string of random digits. The population IDs range from 001 to 600.

Assigning digits to individuals

Since the population size is 600, we need to use three digits for each individual's ID in the random digit string. We will select each set of three consecutive digits as one sample until we have 5 unique samples within the range of 001 to 600.

Selecting the random sample

Starting from the beginning of the string, take each non-overlapping set of three digits and determine if it falls within the range of the population IDs (001-600). If it does, and it hasn't been selected before, it counts as part of the sample.

Listing the sampled IDs

Continue through the string until 5 valid and unique IDs are found. Record these IDs as they are the selected individuals for the sample.

Key concepts

Probability Sampling

Probability sampling is a cornerstone concept in statistics commonly used when conducting research studies to ensure that every member of a population has a known and usually equal chance of being selected for the sample. This method promotes the concept of randomness and non-bias, which are essential for the integrity of the results.

Simple random sampling, the technique mentioned in the exercise, is a basic and widely used type of probability sampling. The beauty of simple random sampling lies in its simplicity and fairness. Every individual, numbered from 001 to 600, could be equally likely to be included in the sample of 5 individuals.

To accomplish this, a random sequence of numbers is generated, and then these numbers are matched up with the population identifiers. This could be done using a random number generator, a lottery system, or even a string of random digits, as shown in the exercise. The sequence provides a non-biased method of selection, ensuring the sample is representative of the whole population, thus allowing for generalizable conclusions to be drawn from the collected data.

Statistical Methods

Statistical methods encompass a broad range of techniques for collecting, analyzing, interpreting, and presenting data. In the context of our exercise, the process falls within the larger framework of statistical sampling which is fundamental for making inferences about a population.

The process starts with understanding your population and defining the sample size. Next, each individual in the population is assigned a unique identifier (in our case, 001 to 600). A random sequence is then used to select members without bias. After securing the sample, statistical analysis can be conducted to estimate characteristics of the entire population, such as mean, variance, or other parameters.

In more complex scenarios, statisticians may employ more sophisticated methods such as stratified sampling, cluster sampling, or systematic sampling, depending on the research objectives and the nature of the population. The proper application of these methods is crucial for achieving precise and valid results.

Survey Sampling Techniques

Survey sampling techniques are strategies used to select a representative subset of individuals from a larger population for the purpose of conducting a survey. These methodologies are designed to provide insights into the opinions, behaviors, or characteristics of the entire group.

An integral part of these techniques is ensuring that the sampling process is free from biases and that the data collected is reliable. Simple random sampling, as seen in our exercise, is just one of the approaches used in surveys.

However, it's important to keep in mind that the process must be carefully managed—it is recommended for larger populations to use random digit tables or software generators to ensure that the randomness is maintained. This precision prevents patterns or clusters, which could influence the results' accuracy, from cropping up. Moreover, clear instructions and a well-understood process are vital to reduce errors. For instance, in our exercise, if the selected number is outside the range of 001-600 or previously selected, it is skipped, which reinforces the randomness and represents the population accurately.