Problem 8 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

In Exercises 5–8, identify the type of sample described. The owner of a community pool wants to ask patrons whether they think the water should be colder. Patrons are divided into four age groups, and a sample is randomly surveyed from each age group.

Step-by-Step Solution

Answer

The type of sample described in the scenario is a stratified sample.

Step 1 - Understanding Basic Sampling Methods

There are different methods of sampling, like simple random sampling, systematic sampling, cluster sampling, and stratified sampling. In stratified sampling, the population is divided into non-overlapping groups or 'strata,' and a simple random sample is selected within each group. Here, the fact that the pool patrons are divided into four age groups suggests a stratified sampling method.

Step 2 - Identifying the Sampling Method

In the given situation, the overall population (the pool patrons) has been divided into non-overlapping groups (specifically, age groups). From each of these groups, a selection of individuals is randomly chosen to respond to the survey. This method of organizing and selecting a sample indicates that a stratified sampling method has been employed.

Key Concepts

Sampling Methods

Understanding different sampling methods is crucial in the field of statistics, particularly when it comes to conducting surveys or experiments. Sampling is the process of selecting a subset of individuals from a population to estimate the characteristics of the whole population.

There are several common types of sampling methods:

• Simple Random Sampling: Every member of the population has an equal chance of being selected.
• Systematic Sampling: Members are chosen from a larger population according to a random starting point and a fixed periodic interval.
• Cluster Sampling: The entire population is divided into groups, or clusters, and a random sample of these clusters is selected.
• Stratified Sampling: The population is divided into subgroups, or strata, and random samples are taken from each stratum.

The choice of sampling method depends on the research objectives, the nature of the population, the resources available, and the level of accuracy required. Stratified sampling, which is the focus of the textbook exercise, ensures representation of all subgroups within the population, which can lead to more accurate and reliable results compared to other sampling methods if the strata are relevant to the subject being studied.

Simple Random Sampling

Simple random sampling is one of the most straightforward and commonly used sampling techniques in statistical analysis. It provides equal chances of inclusion for all members of the population, which helps to minimize biases and enables the results to be more representative of the whole.

In practice, simple random sampling can be performed by using lottery methods, random number tables, or computer-based random number generators. For example, if researchers want to survey college students about their eating habits, they could use a student registry and randomly select students from that list to receive the questionnaire.

Simple random sampling is often used as a component within other complex sampling methods, such as stratified sampling, where each stratum is sampled independently using simple random techniques. When implemented correctly, it ensures that every potential sample has the same probability of being selected, which is a fundamental principle in creating statistically valid samples.

Statistics in Algebra 2

The study of statistics in Algebra 2 involves analyzing data sets to find patterns, make predictions, and evaluate hypotheses. It integrates various statistical measures, probability concepts, and interpretations of numerical data. Students learn about different types of data, how to collect, summarize, interpret, and present it in a meaningful way.

Statistics in Algebra 2 covers topics such as measures of central tendency (mean, median, mode), measures of dispersion (range, variance, standard deviation), and data visualization techniques like histograms, box plots, and scatter plots. Sampling methods, including simple random and stratified sampling, are essential in this context because they are building blocks for understanding how data can be collected in a way that it reflects the population accurately.

Incorporating exercises that involve real-world scenarios, such as the one from the textbook where patrons of different age groups at a community pool are surveyed, helps students grasp the relevance of statistical methods. These applications enhance critical thinking and analytical skills, preparing students for more advanced statistical concepts.