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Worksheet Grade 11 Math

Thornfield's Tap Sampling: Mean Lead Level and Margin of Error

A grade 11 worksheet where students use a fictional city's sample mean and a simulated margin of error to estimate its true citywide tap-water lead level.

Objectives

  1. Students will use a sample mean and a margin of error to build an interval estimate for a population mean.
  2. Students will compare an interval estimate against a regulatory action level to draw a supported conclusion.

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, margin of error, population parameter

Teacher overview

The fictional city of Thornfield samples household taps as part of its Lead and Copper Rule compliance program, and the mean of those sampled taps is only an estimate of the true citywide lead level, an estimate that carries uncertainty. A common misconception is treating a sample mean as if it were the exact population value; this worksheet has students build an honest interval estimate instead, using a margin of error generated by resampling simulation, and then check that interval against the real EPA lead action level.

Thornfield's Tap Sampling: Mean Lead Level and Margin of Error — Grade 11 Math
Name: ____________________________ Date: ______________

Student Worksheet

The fictional city of Thornfield's water utility sampled 25 household taps and found a sample mean lead level of 6.2 ppb. To estimate how much that sample mean might vary from the true citywide mean, the utility's analyst ran a computer simulation that resampled the 25 readings 1,000 times. The middle 95 percent of those 1,000 simulated sample means fell between 5.0 ppb and 7.4 ppb.

Note: under the Lead and Copper Rule, a utility compares its 90th-percentile tap sample, not its mean, with the action level, and the action level is a trigger for action rather than a safe level. EPA's health-based goal for lead is zero. Items 5, 6, and 10 compare the mean with the action level as a practice exercise only.

  1. What sample mean lead level did Thornfield measure across its 25 sampled taps?
  2. The simulated range of sample means runs from 5.0 ppb to 7.4 ppb. Calculate the estimated margin of error (half the width of that range).
  3. Write Thornfield's result as "sample mean ± margin of error."
  4. Calculate the lower bound and the upper bound of Thornfield's interval estimate for the true citywide mean lead level.
  5. The current lead action level is 15 ppb. Is Thornfield's entire interval estimate below that action level?
  6. Starting in 2027, the lead action level drops to 10 ppb. Is Thornfield's entire interval estimate still below that stricter level?
  7. A fictional neighboring city, Millpond, samples only 8 taps and gets a wider simulated range: 3.0 ppb to 11.0 ppb. Calculate Millpond's margin of error.
  8. Compare Thornfield's margin of error to Millpond's. Using the two sample sizes (25 taps versus 8 taps), explain why sampling more taps generally produces a smaller margin of error.
  9. If Thornfield wanted to cut its 1.2 ppb margin of error roughly in half, would it need a larger or smaller sample size for its next round of testing? Explain your reasoning.
  10. Thornfield's next round of sampling produces a new sample mean of 6.8 ppb, with the margin of error holding steady at 1.2 ppb. Write the new interval estimate and state whether it still falls below both the current and 2027 action levels.

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: 6.2 ppb.
    This is the sample mean stated directly in the scenario, calculated from the utility's 25 sampled taps.

  2. Answer: 1.2 ppb.
    Half the width of the 5.0 to 7.4 ppb range: (7.4 − 5.0) ÷ 2 = 2.4 ÷ 2 = 1.2 ppb.

  3. Answer: 6.2; written as sample mean ± margin of error, the result is 6.2 ± 1.2 ppb.
    Written as sample mean ± margin of error, Thornfield's result is 6.2 ± 1.2 ppb: the sample mean (6.2 ppb) plus or minus the margin of error (1.2 ppb), the standard way to report an interval estimate.

  4. Answer: 5.0 ppb to 7.4 ppb.
    Lower bound: 6.2 − 1.2 = 5.0 ppb. Upper bound: 6.2 + 1.2 = 7.4 ppb, which matches the simulated range given. The full interval estimate runs 5.0 to 7.4 ppb.

  5. Answer: yes.
    The interval's upper bound (7.4 ppb) stays below the current 15 ppb action level, so the entire interval is below it.

  6. Answer: yes.
    The interval's upper bound (7.4 ppb) is also below the stricter 10 ppb action level that takes effect in 2027, so the entire interval is also below that level.

  7. Answer: 4.0 ppb.
    Half the width of the 3.0 to 11.0 ppb range: (11.0 − 3.0) ÷ 2 = 8.0 ÷ 2 = 4.0 ppb.

  8. Answer: Millpond's margin of error is larger because it sampled fewer taps.
    Millpond's margin of error (4.0 ppb) is much larger than Thornfield's (1.2 ppb) because Millpond sampled far fewer taps (8 versus 25); a smaller sample gives each simulated resample more room to swing, which widens the range of plausible sample means.

  9. Answer: larger.
    Reducing the margin of error requires a larger sample size; more sampled taps narrow the spread of simulated sample means, which is exactly the pattern seen when comparing Thornfield's 25-tap margin of error to Millpond's 8-tap margin of error.

  10. Answer: 5.6 ppb to 8.0 ppb; yes, below both action levels.
    New interval: 6.8 − 1.2 = 5.6 ppb to 6.8 + 1.2 = 8.0 ppb (upper bound). This interval's upper bound (8.0 ppb) still stays below both the current 15 ppb action level and the 2027 action level of 10 ppb.

Differentiation

Support (below grade level)

Provide a labeled number line showing the sample mean, the margin of error distance on each side, and the resulting lower and upper bounds already drawn for item 4, so students can model items 7 and 10 after that example. Allow calculators for every subtraction and division step.

Extension (above grade level)

Ask students to research, using only the numbers given in this worksheet, what sample size the fictional city of Millpond would likely need to shrink its margin of error down to Thornfield's 1.2 ppb, reasoning from the relationship between sample size and margin of error without a formal formula.

ELL supports

Pre-teach "sample," "margin of error," and "action level" with a simple diagram: a target with a bullseye (the true population mean) and a ring around it (the margin of error) showing where a sample mean is likely to land. Allow students to show item 3 as a labeled number line instead of a written sentence.