Unit 4 — Drawing Inferences about Populations & Probability Models
Description
This unit develops statistical and probabilistic reasoning. Students begin by understanding how random sampling produces representative samples and enables valid inferences about populations. They use data from random samples to estimate population characteristics and assess variation by generating multiple samples of the same size. From there, students compare two populations using measures of center and variability, expressing differences between centers as multiples of a measure of spread. The unit moves to probability, where students understand probability as a number between 0 and 1 expressing likelihood. Students approximate probabilities through experimentation and data collection, recognizing relative frequency. They develop both uniform and non-uniform probability models and use these to find probabilities of events. The unit concludes with compound events, where students represent sample spaces using organized lists, tables, and tree diagrams, and design simulations to estimate probabilities.
Essential Questions
- How do we use samples to draw valid inferences about populations?
- How can measures of center and variability help us compare two groups of data?
- How do we model and predict the likelihood of chance events?
Learning Objectives
- Distinguish between representative and non-representative samples of a population.
- Use random sampling to produce representative samples.
- Draw inferences about a population from a random sample and assess variation through multiple samples.
- Visually compare two distributions with similar variability and express differences between centers as multiples of variability.
- Draw informal comparative inferences about two populations using measures of center and measures of variability.
- Interpret and express the likelihood of a chance event as a number between 0 and 1.
- Approximate the probability of a chance event by collecting data and observing long-run relative frequency.
- Develop uniform probability models by assigning equal probability to all outcomes.
- Develop non-uniform probability models by observing frequencies in data.
- Compare probabilities from a model to observed frequencies and explain discrepancies.
- Represent sample spaces for compound events using organized lists, tables, and tree diagrams.
- Find probabilities of compound events using sample spaces.
- Design and use simulations to estimate probabilities of compound events.
Supplemental Resources
- Dice and spinners for conducting probability experiments and simulations
- Plastic bags or envelopes for storing objects used in chance processes
- Organizing tables and tree diagram templates for representing compound events
- Tally sheets and frequency tables for recording experimental data
- Printed scenario cards for sampling and probability contexts
Statistics and Probability
Students write in science notebooks, construct viable arguments, and critique reasoning of others using mathematical evidence and precise language to communicate mathematical thinking.
Students apply mathematical reasoning to analyze scientific data, represent relationships using equations and graphs, and solve real-world problems involving physical phenomena such as temperature, distance, and population dynamics.
Students use proportional reasoning and quantitative analysis to examine historical economic data, evaluate policy decisions, and assess demographic information across different time periods and societies.
Formative Assessments
- Exit tickets checking understanding of representative samples and probability concepts.
- CPM checkups on 7.SP standards.
- Quizzes on identifying probability ranges and estimating with experimental data.
- Observations of student work on sample space representations and probability calculations.
- Pair-and-share discussions analyzing data from random samples and simulations.
Summative Assessment
Unit 4 test on sampling, inference, and probability; performance assessment designing a sampling study or probability simulation and interpreting results.
Benchmark Assessment
— not configured —
Alternative Assessment
Students may demonstrate understanding through a guided interview or oral explanation of sample representativeness and probability concepts, with visual aids such as labeled diagrams or sample data displays provided. Response options may be narrowed (e.g., selecting from three statements about whether a sample is representative rather than generating an explanation), and extended time or frequent breaks may be provided as needed.
IEP (Individualized Education Program)
Students may benefit from graphic organizers that visually distinguish key statistical concepts such as sample versus population, experimental versus theoretical probability, and compound versus simple events. Providing a reference card with probability vocabulary, model diagrams, and fraction-to-decimal equivalents supports both processing and retention across the unit. Teachers should allow oral or dictated explanations when students are asked to interpret data or justify probabilistic reasoning, reducing the writing demand while still assessing conceptual understanding. Breaking multi-step tasks—such as constructing a sample space or designing a simulation—into clearly numbered, sequential steps with checkpoints helps students manage complexity and monitor their own progress.
Section 504
Students should be provided extended time on quizzes and the unit assessment, particularly for tasks requiring students to construct tree diagrams, organized lists, or written interpretations of probability data. Access to a calculator and a multiplication or fraction reference tool ensures that arithmetic demands do not become a barrier to demonstrating statistical and probabilistic reasoning. Preferential seating and a low-distraction environment are especially beneficial during data collection activities and simulation tasks that require sustained focus and careful record-keeping.
ELL / MLL
Teachers should build and reinforce a visual word wall for this unit that includes terms such as sample, population, probability, outcome, event, and simulation, accompanied by diagrams and numerical examples rather than relying on text-heavy definitions. Directions for tasks involving data collection, sample space construction, or simulation design should be simplified and broken into short steps, with visual models of the expected process or end product provided alongside. Where possible, connecting probability scenarios to contexts familiar across cultural backgrounds—such as games of chance or everyday decision-making—can increase accessibility and engagement with new content.
At Risk (RTI)
Instruction should begin by connecting probability and sampling concepts to students' everyday experiences—such as predicting outcomes of familiar games or making guesses about a school population—to build an accessible entry point before moving to more formal representations. Teachers should offer structured templates for organizing sample spaces and recording experimental data, reducing the cognitive load of format decisions so students can focus on the underlying reasoning. Starting with uniform probability models and simple events before introducing non-uniform models and compound events allows students to build confidence on foundational ideas before encountering greater complexity.
Gifted & Talented
Students who have demonstrated mastery of foundational probability and sampling concepts should be challenged to investigate real-world data sets or research studies, critically evaluating how sampling methods affect the validity of published conclusions. They can extend their understanding of probability models by exploring situations where theoretical and experimental probabilities diverge significantly, constructing arguments about what sample size or simulation design would reduce that gap. Encouraging students to design original simulations for multi-stage compound events—and then analyze the limitations of their models—pushes into deeper statistical thinking and authentic mathematical inquiry.