Making Inferences from Samples
Aligned to S-IC.B.4 — Common Core State Standards for Mathematics.
What this lesson teaches
A well-chosen random sample lets us estimate facts about a whole population, but every estimate has a margin of error reflecting uncertainty.
Worked examples
Example 1
A poll of 1,000 random voters finds 52% support a measure, with a ±3% margin — so true support is likely between 49% and 55%.
Example 2
A quality inspector samples 250 parts at random and finds 6 defective, a rate of 2.4%. The margin of error is roughly 1/√250 = 6.3%, which is wider than the estimate itself — so this sample cannot distinguish a 2% defect rate from a 5% one. To get a margin near 1% the sample must grow to about 10,000.
Example 3
Common mistake: treating a margin of error as covering every kind of error. It measures ONLY the randomness of sampling. It says nothing about a badly worded question, people who refuse to answer, or a sampling frame that leaves out part of the population. A poll conducted only by landline has a printed margin of 3% and could still be off by 10 points, because the missing people are missing systematically.
Practice questions
- Why must a sample be random to be trustworthy?
- If a poll says 60% ±4%, what's the likely range?
- What can bias a survey's results?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Hacer inferencias a partir de muestras
Una muestra aleatoria bien elegida nos permite estimar datos sobre toda una población, pero cada estimación tiene un margen de error que refleja la incertidumbre.
Ejemplo 1: Una encuesta a 1,000 votantes elegidos al azar encuentra que el 52% apoya una medida, con un margen de ±3% — por lo que el apoyo real probablemente está entre el 49% y el 55%.
Ejemplo 2: Un inspector de calidad toma 250 piezas al azar y halla 6 defectuosas, una tasa del 2.4%. El margen de error es aproximadamente 1/√250 = 6.3%, más ancho que la propia estimación — así que esta muestra no puede distinguir una tasa de defectos del 2% de una del 5%. Para lograr un margen cercano al 1%, la muestra debe crecer a unas 10,000 piezas.
Ejemplo 3: Error común: creer que el margen de error cubre todo tipo de error. Mide SOLO la aleatoriedad del muestreo. No dice nada sobre una pregunta mal redactada, sobre la gente que se niega a responder o sobre un marco muestral que deja fuera a parte de la población. Una encuesta hecha solo por teléfono fijo tiene un margen impreso del 3% y aun así puede errar por 10 puntos, porque quienes faltan faltan de forma sistemática.
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