Problem 20 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

A national collegiate athletic association wants to survey 15 of the 120 head football coaches in a division about a proposed rules change. Describe a method for selecting a random sample of coaches to survey.

Step-by-Step Solution

Answer

To create a random sample of coaches, each of the 120 coaches is assigned a unique identifier number from 1 to 120. A random number generator is then used to select 15 different numbers from this range. The coaches corresponding to these 15 numbers form the random sample for the survey.

Step 1: Understand Random Sampling

Random sampling is a procedure for sampling from a population in which (a) the selection of a sample unit is based on chance, and (b) every element of the population has a known, non-zero chance of being selected. Every head football coach should have an equal chance of being selected.

Step 2: Assign a Unique Identifier

Assign each of the 120 head football coaches a unique number from 1 to 120. These numbers will serve as identifiers for the coaches.

Step 3: Use a Random Number Generator

Use a random number generator to select 15 different numbers from the range we created (1 to 120). The app or tool used for this random generation should ensure that the selection of any specific number is independent of the selection of any other number.

Step 4: Select the Coaches

The head football coaches that correspond to the 15 unique numbers chosen by the random number generator will constitute the random sample. Contact these coaches to conduct the survey.

Key Concepts

Understanding Survey Methodology

Survey methodology is all about the techniques and processes used to gather information from a specific group. In the context of the exercise, the objective is to effectively gather opinions from 15 head football coaches out of a total of 120. There are several steps involved in creating a robust survey methodology.

• **Defining the Purpose**: Clearly defining what information needs to be gathered and why it is crucial to guide the process. In our case, it is to collect insights on a proposed rules change.
• **Choosing the Sample**: Since contacting all 120 coaches is impractical, a decision is made to survey a smaller, manageable group of 15.
• **Data Collection**: This involves determining how the information will be collected, for instance, via an online survey, telephone call, or face-to-face meeting.
• **Ensuring Accuracy**: The selection method and survey questions must be designed to avoid biases, ensuring that the resulting data is both reliable and representative of the whole group.

Exploring Population Sampling

Population sampling is a vital part of the survey process. It involves selecting a small group (sample) from a larger pool (population) to infer conclusions about the entire group. Here, the population is 120 head football coaches, and the sample is 15 of them. Let's delve into why sampling is useful.

• **Resource Efficiency**: Collecting data from the entire population can be time-consuming and costly. Sampling offers a practical alternative by focusing efforts on a small, representative portion.
• **Descriptive Representation**: The sample should accurately reflect the diversity of the entire population. This means ensuring every member of the population has a chance of being selected, as seen in our example through random sampling.
• **Margin of Error**: With a proper sampling method, results can be reasonably accurate and applicable to the whole population, even if there is a small margin for error.

How Random Number Generators Aid Sampling

Random Number Generators (RNGs) play an essential role in ensuring a fair sample selection process. They generate numbers in a way where each possible number has an equal chance of appearing. Here's how RNGs make the process seamless:

• **Automatic Selection**: RNGs automate the selection of sample numbers. In our exercise, this involves choosing 15 numbers from 1 to 120 without manual intervention, ensuring impartiality.
• **Uniformity and Independence**: The selection of each number is independent and doesn't influence the choice of other numbers. This ensures every coach has an equal probability of being chosen, which is key in maintaining sample integrity.
• **Versatility**: RNGs can be accessed through various software and applications, making them a flexible tool for any sampling needs.

Using an RNG in the selection of football coaches ensures the survey remains unbiased and representative, ultimately leading to more accurate and reliable results.