Question

A petition with 500 signatures is submitted to a college's student council. The council president would like to determine what proportion of those who signed the petition are actually registered students at the college. There is not enough time to check all 500 names with the registrar, so the council president decides to select a simple random sample of 30 signatures. Describe how she might do this.

Short answer

The president should list all signatories, assign each a unique number within a 1-500 interval, then use a random number generator to pick 30 unique numbers. The individuals assigned those numbers will make up the random sample.

Step-by-step solution

Understand simple random sampling

Simple random sampling is a subset of a statistical population where each member of the subset has an equal probability of being chosen. A simple random sample is supposed to be an unbiased representation of a group.

Create a list

First, the president needs to prepare a comprehensive list of all 500 people who signed the petition.

Assign a unique number

Assign each individual on the list a unique number from 1 to 500.

Use a Random Number Generator

Then use a random number generator to generate 30 distinct numbers within this range. The generator should be set to not repeat numbers and only generate numbers within the 1-500 interval.

Select the Sample

Those individuals whose corresponding numbers were generated forms the simple random sample of 30 students whose signatures she checks with the registrar. This creates an equal likelihood of being selected for each of the petitioners, achieving a simple, random sample

Key concepts

Statistical Population

When it comes to understanding statistical population, it's like considering everyone at a big party. Imagine you're at the party and want to know how many people prefer chocolate cake over vanilla. The entire group of people at the party represents the statistical population. In the context of the student council president's task, the 500 signatures on the petition form the population - that is, all the individuals who could potentially be examined to answer her question about registered student status. Establishing an accurate population is crucial since it's the basis from which samples are drawn.
It's akin to knowing you have both chocolate and vanilla slices to serve before you can ask each guest their preference. Similarly, you need a complete list of all petitioners before you draw the sample.

Random Number Generator

Using a random number generator is like pulling raffle tickets from a hat. You want to give everyone an equal shot at winning, without any peeking! A random number generator is a tool that can pick numbers without any pattern or bias. In the scenario with our council president, after assigning each signature a unique number from 1 to 500, she'd use a random number generator to randomly select 30 unique numbers.
To ensure fairness, imagine if our raffle was rigged or the same numbers kept getting picked – it wouldn't be fair, right? That's why the random number generator must be set to not repeat numbers and stay within the limits (1 to 500), ensuring each signature has an equal chance of being chosen.

Sampling Methods

Imagine shopping for fruit and wanting a taste of the available mangoes. You randomly pick a few to try instead of tasting them all. This is a bit like sampling methods: the ways in which we can pick 'tastes' or samples from our bigger group - the population. Simple random sampling is one type, like selecting mangoes blindly from a basket.
There are others, too, such as stratified sampling, where you first divide the population into groups (like separating ripe and unripe mangoes) and then randomly sample within each group. Or cluster sampling, where you'd randomly pick whole boxes of mangoes rather than individual fruits. The choice of sampling method depends on the goals of your study and the nature of your population.

Unbiased Sample Representation

An unbiased sample representation is the golden ticket of sampling; it's what makes the results trustworthy. Think of it as a mirror reflecting the full variety of a crowd at a festival. If the mirror only shows people wearing blue hats, you'd get the wrong idea about what everyone else likes to wear.
An unbiased sample does its best to accurately reflect everyone in the population. It's like making sure that, in our earlier party scenario, both chocolate and vanilla cake fans are randomly asked about their preferences without leaning towards one group. The council president's sample of 30 signatures should ideally mirror the larger group of 500, giving a fair representation of the registered student status within the entire group.