Opening
A town asks a hundred people what they think and announces what ten thousand people believe. That is quite an impressive promotion for a hundred people.
Teaching narrative
Statistics is not fundamentally a collection of arithmetic tricks. It is a disciplined way to learn from observations when the complete answer is not available. Begin by asking which people, events, or things constitute the population, and which were actually measured.
Now separate the population parameter from the sample statistic. The first is the quantity we want to learn; the second is calculated from what we actually observed. The sample statistic is not magically identical to the parameter merely because a spreadsheet printed it with four decimal places.
We also need to distinguish collecting observations from assigning interventions. Randomly selecting survey respondents can help us generalize to a population. Randomly assigning a treatment can help separate causes from preexisting differences. Those operations solve different problems.
Before we leave, write down a question, an observational unit, a measurement, a sample-selection mechanism, and a claim the design cannot establish. You have now done more honest statistical work than many pages of equations ever accomplish.
Before the experiment
Would surveying two hundred enthusiastic volunteers produce the same inferential evidence as randomly selecting two hundred enrolled students?
Live demonstration
Show a hypothetical university with 20,000 students. Select 200 for an imagined survey and ask which number is the population size, which is the sample size, and what study time means.
Exit question
One sentence: what could the statistic describe accurately even if it did not represent the population?