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Lesson plan Grade 12 Math

Copper Bend Tap Sampling: Estimating Lead Levels with Margin of Error

A grade 12 statistics lesson where students estimate a mean lead level from simulated tap samples and build a margin of error for a fictional water system.

Objectives

  1. Students will calculate the standard error of a sample mean from a sample size and standard deviation.
  2. Students will build a margin of error and confidence interval for a sample mean and compare it to a real regulatory threshold.

Vocabulary

Action level
A trigger number in lead and copper testing. If too many homes test above this level, the utility must take steps like improving treatment and telling the public.

Also: sample mean, standard error, margin of error, confidence interval

Teacher overview

Copper Bend Water Utility (a fictional utility used for this lesson) cannot test every tap in its system, so for a screening survey it tests a random sample of household taps and uses that sample to estimate the average lead level across the whole system. Official Lead and Copper Rule compliance sampling works differently: it targets the homes most likely to have lead and compares the 90th percentile result, not the mean, with the action level. The current federal lead action level is 15 ppb, dropping to 10 ppb in 2027 under the Lead and Copper Rule Improvements, so knowing not just a sample's average but how much that average might vary from the true system-wide level helps a utility judge how close its lead levels may be to that threshold.

A common misconception is treating a sample mean as if it were the exact, error-free truth about the whole system, rather than an estimate with some uncertainty attached. This lesson has students calculate that uncertainty directly, using the standard error of the mean and a margin of error, so that "our sample averaged 6.2 ppb" becomes the more honest and more useful statement, "our sample suggests the true system mean falls somewhere between about 4.8 and 7.6 ppb."

Procedure

Hook (5 min): Ask students: if a utility tests 25 taps out of 10,000 homes and gets an average lead reading, how confident should anyone be that this average represents every home in the system? Collect initial reactions without confirming an answer.

Direct instruction (15 min): Introduce sampling as inference: a random sample of taps stands in for the whole distribution system, the same logic behind any survey that uses a sample to describe a whole system. Introduce the standard error formula, SE = s ÷ √n, where s is the sample's standard deviation and n is the sample size, and a commonly used approximate 95 percent margin of error, ME ≈ 2 × SE. Work through Copper Bend's own fictional sample together: 25 taps tested, sample mean 6.2 ppb, sample standard deviation 3.5 ppb. Calculate SE = 3.5 ÷ √25 = 0.7 ppb, ME ≈ 2 × 0.7 = 1.4 ppb, and the resulting confidence interval, 6.2 ± 1.4, or 4.8 to 7.6 ppb.

Guided practice (15 min): In pairs, students calculate the same statistics for a second Copper Bend neighborhood's tap sample: 16 taps tested, sample mean 9.0 ppb, sample standard deviation 4.0 ppb. Pairs find SE = 4.0 ÷ √16 = 1.0 ppb, ME ≈ 2.0 ppb, and a confidence interval of 7.0 to 11.0 ppb, then compare that interval to both the current 15 ppb action level and the 2027 action level of 10 ppb.

Independent practice (10 min): Each student answers, in writing: (1) what is the standard error for the original Copper Bend sample, using the given n = 25 and s = 3.5 ppb; (2) what is the resulting 95 percent margin of error and confidence interval for Copper Bend's mean lead level; (3) what is the standard error, margin of error, and confidence interval for the second neighborhood's sample (n = 16, mean 9.0 ppb, s = 4.0 ppb); and (4) does the second neighborhood's confidence interval raise concern under the current 15 ppb action level, the future 10 ppb action level, both, or neither.

Closure / exit ticket (5 min): Students answer one question on an index card: "If Copper Bend had tested only 9 taps instead of 25, using the same standard deviation of 3.5 ppb, would the margin of error be larger or smaller than 1.4 ppb? Explain using the standard error formula." Collect and scan for correct reasoning (a smaller sample size increases standard error and margin of error) before dismissal.

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

Answer Key

  1. Answer: 0.7 ppb.
    SE = s ÷ √n = 3.5 ÷ √25 = 3.5 ÷ 5 = 0.7 ppb.

  2. Answer: 4.8 to 7.6 ppb, centered at 6.2 ppb.
    ME ≈ 2 × 0.7 = 1.4 ppb. Confidence interval = 6.2 ± 1.4 = 4.8 to 7.6 ppb. Both bounds fall well below the current 15 ppb action level and the future 10 ppb action level.

  3. Answer: 7.0 to 11.0 ppb, centered at 9.0 ppb.
    SE = 4.0 ÷ √16 = 4.0 ÷ 4 = 1.0 ppb. ME ≈ 2 × 1.0 = 2.0 ppb. Confidence interval = 9.0 ± 2.0 = 7.0 to 11.0 ppb.

  4. Answer: Concern under the 2027 action level only.
    The interval's upper bound of 11.0 ppb stays below the current 15 ppb action level, so this neighborhood's average stays below today's 15 ppb level; however, 11.0 ppb exceeds the 10 ppb action level that takes effect in 2027, so the same sample would raise concern once the stricter standard applies. Because the action level is applied to the 90th percentile of targeted samples rather than to a mean, this comparison is a screening signal for further sampling, not a compliance determination.

Exit ticket answer: Larger.
Standard error is calculated as s ÷ √n, so a smaller sample size (9 instead of 25) produces a smaller square root in the denominator, which makes the standard error, and therefore the margin of error, larger: SE = 3.5 ÷ √9 = 3.5 ÷ 3 ≈ 1.17 ppb, giving a margin of error of about 2.33 ppb, noticeably larger than the original 1.4 ppb.

Differentiation

Support (below grade level)

Provide the standard error and margin of error formulas on a printed reference card with the Copper Bend example already worked through step by step, so students can follow the same steps with the second neighborhood's numbers rather than deriving the process from scratch. Allow calculators for the square root steps.

Extension (above grade level)

Ask students to determine how large a sample size the fictional Copper Bend utility would need, keeping the same standard deviation of 3.5 ppb, to cut the margin of error in half, from 1.4 ppb to 0.7 ppb, and to explain in general terms why cutting a margin of error in half requires roughly quadrupling the sample size rather than just doubling it.

ELL supports

Pre-teach "sample," "mean," "margin of error," and "confidence interval" using a simple visual of a target with an estimate as an arrow and a shaded band around it representing the range of likely true values. Provide the formulas with each symbol labeled in plain language (s = spread of the data, n = number of taps tested) next to the mathematical notation.