Problem 6 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

In Exercises 5–8, identify the type of sample described. Each employee in a company writes their name on a card and places it in a hat. The employees whose names are on the first two cards drawn each win a gift card.

Step-by-Step Solution

Answer

The type of sample described in the situation is a Simple Random Sample.

Step 1: Understand Sample Types

There are different types of samples in statistics: Simple Random Sample, Systematic Sample, Convenience Sample, Cluster Sample, and Stratified Sample.

Step 2: Analyze the Given Situation

In this case, each employee writes their name on a card and puts it in a hat. The employees whose names are drawn from the hat are considered as the sample. This is a case of random selection as any employee has an equal and independent chance of being chosen.

Step 3: Identify the Type of Sample

From the analysis, it's clear that the selection process is random where all employees have equal chances. This aligns with the definition of a Simple Random Sample, where every member of a population has an equal chance to be included in the sample and each selection is made independently of the others.

Key Concepts

Simple Random Sample

Understanding the concept of a Simple Random Sample is fundamental to grasping many principles of statistics. A Simple Random Sample (SRS) refers to a sampling method where every member of a population has an equal probability of being selected. This method is the gold standard for statistical sampling because it aims to reduce sampling bias and ensures each member is chosen without influence from external factors.

In the exercise provided, employees place their names in a hat, and a draw determines who receives a gift card. This exemplifies a Simple Random Sample, as each employee's chance of being selected is equal. An essential feature of an SRS is independence; this means the selection of one individual does not affect the selection of another. Practically speaking, this is like drawing names from a hat without replacement, ensuring that once someone is chosen, they cannot be selected again for that particular sample.

It is crucial to ensure true randomness in this method. Real-life implementations could utilize various mechanisms, such as random number generators or lottery systems, to simulate the 'hat and draw' approach in larger populations. By ensuring each individual in the population has an equal chance of being selected, this method provides a representative sample of the population, which is vital for unbiased statistical analysis.

Random Selection

Essence of Random Selection

Random selection is a cornerstone concept in statistical sampling, and it serves as the basis for creating samples that accurately reflect the population. The term refers to the process of choosing individuals or items in such a way that every possible selection has a known and equal chance of occurring. A critical aspect is that it must be free from any system or pattern that could influence the outcome.

The exercise illustrates a practical application of random selection, wherein the 'luck of the draw' is the sole determinant of who wins the gift card. In statistical terms, this process ensures that the sample collected doesn't favor any segment of the population over another. Why is this important? Because it reduces the potential for biases that could skew results and compromise the validity of any conclusions drawn from the data collected.

Random Selection in Practice

It's important to note that random selection doesn't just happen on a small scale, like in the exercise. In larger surveys and studies, statisticians must use technological tools or methodologies (like random digit dialing or randomized algorithms) to maintain this level of fairness. Whether in an office raffle or a massive public health survey, the integrity of random selection is the linchpin that upholds the reliability of statistical outcomes.

Statistical Sampling Methods

In statistics, there are multiple sampling methods, each with its own significance and appropriate context of use. Apart from a Simple Random Sample, which we have discussed, there are several other methods: Systematic, Convenience, Cluster, and Stratified sampling. Understanding when and how to use these methods is essential for accurate data collection and analysis.

Variety of Sampling Methods

A Systematic Sample is one where you select every 'n-th' individual from the population, which can be effective when dealing with large datasets. Convenience Sampling involves selecting individuals most accessible to the researcher, often used in preliminary research where precision is not the primary goal. Cluster Sampling divides the population into clusters, then randomly selects entire clusters, making it more practical for geographically dispersed populations. Lastly, Stratified Sampling requires dividing the population into strata, usually based on a characteristic, and then taking proportionate samples from each stratum.

Each method has its strengths and weaknesses, and the choice among them should be dictated by the research objectives, resources, and the nature of the population being studied. An understanding of these methods and their proper application is key to conducting robust research that yields reliable insights into the population or phenomena of interest.