Math problem set Grade 11 Science
Winterset Well 6: Fitting a Regression Line to a Nitrate Trend
Grade 11 students fit a regression line to fictional groundwater nitrate readings, predict a future value, and compute a correlation coefficient.
Objectives
- Students will calculate the mean and deviations needed to fit a least-squares regression line to a two-variable data set.
- Students will use a regression equation to predict a future value and evaluate a residual.
- Students will compute and interpret a correlation coefficient.
Vocabulary
Key words: regression line, residual, correlation coefficient, slope, intercept
Teacher overview
Winterset (a fictional town created for this problem set) monitors a groundwater well, called Well 6, near farmland every two years for nitrate-nitrogen, and this problem set uses five of those readings to fit a least-squares regression line and judge how well it predicts the trend. A common misconception this math is designed to correct: students often treat a correlation coefficient close to 1 as proof that every data point sits exactly on the line, when even a very strong linear relationship, like Well 6's, still has small nonzero residuals at nearly every point.
Problems
Well 6's nitrate-nitrogen readings (a fictional data set created for this problem set) are shown below, with "Years since 2016" as the x-variable and "Nitrate-N (mg/L)" as the y-variable.
| Year | Years since 2016 (x) | Nitrate-N, mg/L (y) |
|---|---|---|
| 2016 | 0 | 3.0 |
| 2018 | 2 | 4.0 |
| 2020 | 4 | 5.5 |
| 2022 | 6 | 7.0 |
| 2024 | 8 | 8.0 |
- Calculate the mean of the x-values (x̄) and the mean of the y-values (ȳ).
- Complete a deviation table: for each of the five years, calculate (x − x̄) and (y − ȳ).
- Using your deviations from item 2, calculate Σ(x − x̄)(y − ȳ) and Σ(x − x̄)².
- Calculate the slope of the least-squares regression line, b = Σ(x − x̄)(y − ȳ) ÷ Σ(x − x̄)², and state what it means in mg/L per year.
- Calculate the intercept, a = ȳ − b·x̄, and write the full regression equation in the form y = a + bx.
- Use your equation from item 5 to predict Well 6's nitrate-N level in 2030 (x = 14, since 2030 is 14 years after 2016).
- Suppose local regulations cap nitrate-nitrogen at 10 mg/L for wells in this scenario. Using your equation, find the x-value where the predicted nitrate-N level first reaches 10 mg/L, then state the first even sampling year (2016, 2018, 2020, and so on) at which the predicted value is at or above 10 mg/L.
- Calculate the residual (actual − predicted) for the 2022 sampling year, and explain in one sentence what a residual of that size tells you about how well the line fits the data at that point.
- Calculate the correlation coefficient, r = Σ(x − x̄)(y − ȳ) ÷ √(Σ(x − x̄)² × Σ(y − ȳ)²), given that Σ(y − ȳ)² = 17 for this data set. Round to three decimal places and describe the strength of the linear relationship it shows.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
- Answer: x̄ = 4, ȳ = 5.5 mg/L. x̄ = (0+2+4+6+8) ÷ 5 = 20 ÷ 5 = 4. ȳ = (3.0+4.0+5.5+7.0+8.0) ÷ 5 = 27.5 ÷ 5 = 5.5.
- Answer: listed in order from 2016 to 2024. (x − x̄): −4, −2, 0, 2, 4. (y − ȳ): −2.5, −1.5, 0, 1.5, 2.5.
- Answer: Σ(x − x̄)(y − ȳ) = 26; Σ(x − x̄)² = 40. Products: (−4)(−2.5)=10, (−2)(−1.5)=3, (0)(0)=0, (2)(1.5)=3, (4)(2.5)=10; sum = 26. Squares: 16, 4, 0, 4, 16; sum = 40.
- Answer: 0.65 mg/L per year. b = 26 ÷ 40 = 0.65, meaning Well 6's nitrate-N level rises by about 0.65 mg/L for each additional year.
- Answer: y = 2.9 + 0.65x. a = 5.5 − (0.65 × 4) = 5.5 − 2.6 = 2.9.
- Answer: 12.0 mg/L. y = 2.9 + 0.65(14) = 2.9 + 9.1 = 12.0 mg/L.
- Answer: 2028. Solving 10 = 2.9 + 0.65x gives x = 7.1 ÷ 0.65 ≈ 10.92, which falls between the 2026 sample (x=10: y = 2.9+6.5 = 9.4 mg/L, below 10) and the 2028 sample (x=12: y = 2.9+7.8 = 10.7 mg/L, at or above 10), so 2028 is the first even sampling year the predicted value reaches 10 mg/L.
- Answer: 0.2 mg/L. Predicted value at x=6: y = 2.9 + 0.65(6) = 2.9+3.9 = 6.8 mg/L. Residual = actual − predicted = 7.0 − 6.8 = 0.2 mg/L, a small positive residual showing the actual 2022 reading sat only slightly above the line's prediction, so the line fits closely at that point.
- Answer: 0.997. r = 26 ÷ √(40 × 17) = 26 ÷ √680 ≈ 26 ÷ 26.077 ≈ 0.997, which rounds to 0.997 and shows a very strong positive linear relationship between year and nitrate-N level.
Differentiation
Support (below grade level)
Provide the deviation table from item 2 already completed, so students begin at item 3 with the values ready to sum. Provide the regression-line and correlation-coefficient formulas printed with each symbol labeled in plain language.
Extension (above grade level)
Ask students to calculate r² from their answer to item 9 and explain, in their own words, what percent of the variation in nitrate-N level the regression line accounts for, then connect that number back to the small residual they found in item 8.
ELL supports
Pre-teach "regression," "residual," and "correlation" using a simple scatter plot with a drawn line and one point marked to show the gap (residual) between the point and the line. Provide the formulas as fill-in templates where students substitute numbers into labeled blanks rather than recalling the symbols from memory.