Question

Of the 6500 students enrolled at a community college, 3000 are part time and the other 3500 are full time. The college can provide a list of students that is sorted so that all full-time students are listed first, followed by the part-time students. a. Select a stratified random sample that uses full-time and part-time students as the two strata and that includes 10 students from each stratum. Describe the procedure you used to select the sample, and identify the students included in your sample by placement on the sorted list. b. Does every student at this community college have the same chance of being selected for inclusion in the sample? Explain.

Short answer

A stratified random sample is selected by numbering each full-time and part-time student and then randomly selecting 10 students from each group using a random number generator. As there are different amounts of full-time and part-time students, not every student has the same chance of being included in the sample.

Step-by-step solution

Understand stratified random sampling

Stratified random sampling is a method of sampling that includes the division of a population into smaller sub-groups known as strata. In stratified random sampling, the strata are formed based on shared attributes or characteristics. In our case, the students of this community college form the population. They can be divided into two strata: full-time students and part-time students. A stratified random sample is then selected by independently sampling from each stratum.

Select the samples from each stratum

The community college has 3500 full-time students and 3000 part-time students, and it is required to select 10 students randomly from each stratum. This can be achieved by numbering each full-time student from 1 to 3500 and each part-time student from 3501 to 6500. A random number generating tool can then be used to select 10 random numbers for each stratum. The students corresponding to these numbers in the sorted list would form our sample.

Answer the question of equal opportunity

Given that there are 3500 full-time students and 3000 part-time students, and that the same number of students (10) is selected from each stratum, a full-time student has a 10 out of 3500 chance of getting selected, while a part-time student has a 10 out of 3000 chance of getting selected. Therefore, not every student at this community college has the same chance of being selected for inclusion in the sample.

Key concepts

Statistics Education

Understanding the fundamentals of statistics is crucial for navigating the data-driven world we live in. A key component of statistics education is learning about various sampling methods, including stratified random sampling, which ensures that subgroups within a population are adequately represented. This approach can improve the accuracy and reliability of conclusions drawn from data analysis. To educate students effectively, it's imperative to break down complex terms and methodologies into digestible parts, leading to a more thorough comprehension of the discipline. Moreover, integrating real-world examples, like the college's enrollment data, aids in visualizing abstract concepts and fosters a practical application of statistical knowledge.

Sampling Methods

Sampling methods are the techniques used to select a part of a population for study. Stratified random sampling, the focus of our exercise, is particularly effective when the population contains several distinct subgroups. In this method, the population is divided into strata, and samples are taken from each stratum. This ensures that each subgroup is represented in the final sample, contributing to the precision of the results. It's important to recognize that this method can be contrasted with simple random sampling, where each member of the population has an equal chance to be included in the sample. Both methods provide valuable insights but are chosen based on the research needs and population characteristics.

Improving Stratified Sampling

To improve the stratified random sampling process, we can ensure that the size of the sample from each stratum is proportional to the stratum's size in the population. This is not used in the college example but can be pivotal in obtaining representative insights.

Probability

Probability is the measure of the likelihood that an event will occur. It is a fundamental concept in statistics that helps in predicting outcomes and making decisions based on incomplete information. In the context of stratified random sampling, probability enables us to assess whether every individual has an equal chance of being selected. As seen in the exercise, different probabilities for different strata imply that not all students had an equal chance to be a part of the sample. Understanding these probabilities allows statisticians to evaluate the fairness and bias in their sampling process, and how this may affect the generalizability of their findings to the entire population.

Data Analysis

Data analysis is the process of inspecting, cleaning, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. In our sampling method example, data analysis begins once the stratified sample is collected. It involves assessing the representativeness of the sample, comparing subgroups, and, if necessary, adjusting for any biases. For students, mastering data analysis techniques is vital to accurately interpret the information that samples reveal about larger populations. When educating on data analysis, it is essential to emphasize critical thinking and the ability to challenge results, especially where sampling bias may have skewed data.