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Math problem set Grade 12 Science

Larkspur Bend pH Sampling: Estimating a Mean with Margin of Error

A grade 12 statistics problem set where students estimate a fictional system's mean pH, apply a margin of error, and judge whether the system meets the EPA pH standard.

Objectives

  1. Students will calculate the mean, range, and sample standard deviation of a pH sampling data set.
  2. Students will apply a stated margin of error to build a confidence interval around a sample mean.
  3. Students will judge whether a confidence interval falls within a water-quality standard's range.

Vocabulary

Key words: margin of error, confidence interval, sample mean, standard deviation

Teacher overview

Larkspur Bend, a fictional water system used for this problem set, samples pH at ten sites across its distribution network each month and tracks a single system-wide estimate. EPA does not enforce its secondary standards; they are guidelines for taste, odor, and pipe corrosion, although some states adopt them. Because that estimate comes from a sample rather than a measurement of every drop of water in the system, it carries some uncertainty, which is why this set pairs the mean with a margin of error rather than treating the sample mean as an exact value. Students calculate that mean and its spread from a month of sample data, then use a stated margin of error to judge whether the system meets the EPA's secondary pH standard of 6.5 to 8.5.

A common misconception in this math is treating a single sample mean as if it were the true system-wide value with certainty. The margin of error tells us how much that estimate could reasonably vary if the sampling were repeated, which is why the second month's result in problems 6 and 7 is borderline rather than a clear pass or fail.

Larkspur Bend pH Sampling: Estimating a Mean with Margin of Error — Grade 12 Science
Name: ____________________________ Date: ______________

Problems

Larkspur Bend's ten site readings for this month are: 7.2, 7.5, 6.9, 7.8, 7.1, 7.4, 7.6, 7.0, 7.3, and 7.7. Use these readings for problems 1 through 5.

  1. Calculate the sample mean of the ten pH readings.

  2. Calculate the range of the ten readings (the highest value minus the lowest value).

  3. Calculate the sample standard deviation of the ten readings. Show your deviations from the mean, their squares, the sum of the squares, and the final square root. (Divide the sum of squared deviations by 9, one less than the sample size, before taking the square root.)

  4. A repeated-sampling simulation of Larkspur Bend's monitoring network produced a margin of error of ±0.15 pH units for a sample of this size. Using your mean from problem 1, build the confidence interval (mean minus the margin, mean plus the margin).

  5. The EPA's secondary standard for drinking water pH is 6.5 to 8.5. Does the confidence interval from problem 4 fall entirely within that range? Explain what your answer tells Larkspur Bend about whether it met the guideline this month.

  6. A different month's sampling at the same ten sites produces a mean of 8.42 pH, with the same ±0.15 margin of error. Build the confidence interval for this second month.

  7. Does the confidence interval from problem 6 fall entirely within the EPA's 6.5 to 8.5 standard range? Based on your answer, what should Larkspur Bend's operators consider doing?

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: 7.35 pH.
    Sum = 7.2 + 7.5 + 6.9 + 7.8 + 7.1 + 7.4 + 7.6 + 7.0 + 7.3 + 7.7 = 73.5. Mean = 73.5 / 10 = 7.35.

  2. Answer: 0.9 pH.
    Range = highest (7.8) - lowest (6.9) = 0.9.

  3. Answer: about 0.30 pH.
    Deviations from the mean of 7.35 are: -0.15, 0.15, -0.45, 0.45, -0.25, 0.05, 0.25, -0.35, -0.05, 0.35. Their squares are: 0.0225, 0.0225, 0.2025, 0.2025, 0.0625, 0.0025, 0.0625, 0.1225, 0.0025, 0.1225, which sum to 0.825. Dividing by 9 gives a variance of about 0.0917, and the square root of 0.0917 is about 0.30.

  4. Answer: [7.20, 7.50]
    7.35 - 0.15 = 7.20 (lower bound); 7.35 + 0.15 = 7.50 (upper bound).

  5. Answer: Yes, entirely within the standard.
    Both 7.20 and 7.50 fall inside the 6.5 to 8.5 range, so the entire confidence interval sits within the EPA secondary standard, meaning Larkspur Bend can be confident this month's system-wide pH meets the guideline, with room to spare on both ends.

  6. Answer: [8.27, 8.57]
    8.42 - 0.15 = 8.27 (lower bound); 8.42 + 0.15 = 8.57 (upper bound).

  7. Answer: No, not entirely within the standard.
    The upper bound of 8.57 exceeds the 8.5 limit, even though the sample mean of 8.42 itself is under it. Because the confidence interval straddles the standard's upper edge, operators should treat this result as a caution rather than a clear pass, investigate the cause of the higher readings, and consider adjusting treatment before the next sampling round.

Differentiation

Support (below grade level)

Provide a printed table with columns for value, deviation, and squared deviation already labeled, so students only need to fill in numbers for problem 3 rather than design the table themselves. Allow a calculator for every step.

Extension (above grade level)

Ask students to find how large the margin of error would need to shrink, holding the problem 6 mean of 8.42 fixed, for the entire confidence interval to fall back within the 6.5 to 8.5 standard. Have them set up and solve the inequality for the unknown margin.

ELL supports

Pre-teach "margin of error" and "confidence interval" with a simple number-line diagram showing a single point (the mean) surrounded by a shaded band (the margin) on either side. Allow problems 5 and 7 to be answered as "yes/no plus one reason" rather than full paragraphs.