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Classroom activity Grade 12 Math

Alder Basin Watershed: A Contamination Sampling Simulation

A grade 12 statistics activity where students draw simulated samples from a fictional watershed to test whether a contamination model matches observed data.

Objectives

  1. Students will simulate random sampling from a population with a known proportion and record the results.
  2. Students will compare a simulated sample result to a claimed population model to judge whether the model is consistent with the data.

Vocabulary

Key words: random sample, sample proportion, sampling variation, simulation

Teacher overview

Alder Basin Watershed (a fictional watershed used for this activity) has a large network of monitoring sites, and the regional water authority claims that about 20 percent of these sites exceed a screening threshold for a common contaminant. This activity turns that claim into a physical simulation: each token bag represents the watershed's sites in the exact 20-percent proportion, and each group's draw of 20 tokens represents a real field sampling event. Every red token drawn stands for a site that exceeds the threshold, matching the physical drawing directly to the statistical act of random sampling.

Because sampling always involves some random variation, no single group's sample count will match the claimed 20 percent exactly. The activity's purpose is to build, from the whole class's combined results, a sense of how much variation is normal under the claimed model, so that an unusually high count in a real watershed sample can be recognized as a warning sign rather than dismissed as ordinary noise.

Activity Directions

  1. Before class, prepare one bag per group (or one shared bag used carefully between groups) containing 100 tokens: 20 red tokens, representing sites that exceed the screening threshold, and 80 blue tokens, representing sites that do not. This 20-to-80 ratio represents the water authority's claim that about 20 percent of Alder Basin's monitoring sites exceed the threshold.

  2. Explain to the class that each group will simulate a field sampling event by drawing 20 tokens from the bag, one at a time, recording the color of each draw, and returning the token to the bag and remixing before the next draw. Drawing with replacement keeps each draw's chance of a red token at exactly 20 percent, matching random sampling from a much larger population.

  3. Divide the class into 8 to 10 small groups of 2 or 3 students each. Give each group a bag and a tally sheet with two columns, one for red draws and one for blue draws.

  4. Each group completes its 20 draws, remixing the bag after every single draw, and records a tally mark in the correct column for each one.

  5. Each group calculates its own sample proportion, the number of red draws divided by 20, and reports both its raw count of red draws and its proportion to a class recorder, who lists every group's count on the board or a shared chart.

  6. As a class, build a simple tally chart or dot plot of all groups' red-token counts (out of 20 draws each). Mark the model's expected count, 4 red draws, since 20 percent of 20 equals 4, directly on the chart.

  7. This expected count of 4 red draws out of 20 total draws corresponds to an expected sample proportion of 4 ÷ 20 = 0.20, or 20 percent. Mark this expected proportion on the chart next to the expected count.

  8. Lead a brief discussion of the chart: which counts appeared most often across the groups, and which counts, if any, appeared far from the expected count of 4. Use the discussion to describe, in the class's own words, roughly which counts feel like ordinary sampling variation and which would feel surprising if the 20 percent claim is true.

  9. End the activity once every group has reported its count and the class chart is complete. Gather students for the debrief using the analysis questions below.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key

The items below are numbered to match the step in Activity Directions where each answer is found or discussed (steps 1 through 5 and 8 through 9 are setup, data collection, and wrap-up with no single numeric answer of their own); items 10 through 12 are analysis questions for the debrief that follows step 9.

  1. Answer: 4 red draws.
    If the claimed 20 percent model is exactly correct, the expected number of red draws out of a sample of 20 is 0.20 × 20 = 4.

  2. Answer: 0.20, or 20 percent.
    The model's expected count of 4 red draws out of 20 total draws corresponds to a sample proportion of 4 ÷ 20 = 0.20, or 20 percent, matching the claimed population proportion exactly.

Debrief questions

  1. Debrief question: A group reports 11 red draws out of 20. Is that count inside or outside the typical range of results expected under the 20 percent model?
    Answer: Outside the typical range.
    Ordinary sampling variation under a correct 20 percent model typically produces red counts of roughly 1 to 7 out of 20 draws in a class-sized set of simulations. A count of 11 red draws sits well above that typical range, so it would stand out as unusual rather than blending in with the class's other results.

  2. Debrief question: Is that group's result more consistent with (a) ordinary sampling variation around a true 20 percent rate, or (b) a true contamination rate higher than 20 percent?
    Answer: Option (b), a true rate higher than 20 percent, is more consistent with a count this far outside the expected range.
    A single sample can occasionally land outside the typical range purely by chance, but a result as far outside that range as 11 out of 20 is unlikely to happen from ordinary sampling variation alone if the true rate really is 20 percent, so it is more consistent with the watershed's true contamination rate actually being higher than the water authority's claimed 20 percent.

  3. Debrief question: If the claimed rate were 5 percent instead of 20 percent, would you expect the simulated counts to cluster more tightly (narrower) or more loosely (wider) around their expected count?
    Answer: Narrower.
    A model with a lower claimed proportion, such as 5 percent, produces a smaller amount of sampling variation than a 20 percent model does, because the spread of a sample proportion depends on p times (1 minus p), which is smaller when p is close to 0 or 1 than when p is closer to 0.5. A 5 percent simulation's counts out of 20 draws would cluster more tightly around its expected count of 1 than the 20 percent simulation's counts cluster around 4.

Differentiation

Support (below grade level)

Reduce the sample size to 10 draws per group instead of 20 (with an expected count of 2 red draws), and provide a pre-printed tally sheet with the two columns already labeled and the expected count already marked, so students can focus on the physical drawing and counting process.

Extension (above grade level)

Ask students to calculate the standard error of the sample proportion for the 20-draw simulation using SE = the square root of [p(1 − p) ÷ n] with p = 0.20 and n = 20, then use that standard error to explain, in numeric terms rather than just qualitatively, why a count of 11 red draws is roughly four standard errors away from the expected count of 4.

ELL supports

Pre-teach "random sample," "proportion," and "simulation" using a simple diagram of tokens being drawn from a bag with an arrow pointing to a tally sheet. Allow groups to report their results using the tally marks and numbers on their sheet directly, without needing to explain the process in a full sentence during the reporting step.