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Math problem set Grade 11 Math

Tolliver Creek Road Salt: Chloride and Conductance Regression

A grade 11 problem set fitting a regression line to paired conductance and chloride data from a fictional creek, then predicting, checking residuals, and reading r.

Objectives

  1. Students will use a fitted regression line to predict a value and compute residuals from paired water data.
  2. Students will interpret the slope, intercept, and correlation coefficient of a monitoring data set in context.

Vocabulary

Key words: regression line, residual, correlation coefficient, specific conductance, chloride

Teacher overview

When road salt washes into a stream, it raises both the chloride concentration and the specific conductance, so field crews often measure the easy quantity, conductance, and use a fitted line to estimate the harder one, chloride. This set gives students a small fictional data set from Tolliver Creek, a least-squares line already fitted with technology, and the correlation coefficient, then asks them to predict, check residuals, and interpret. A common misconception it corrects is reading the y-intercept as a real chloride value; here the intercept is negative, a reminder that a fitted line should not be trusted far outside the data it came from.

Tolliver Creek Road Salt: Chloride and Conductance Regression — Grade 11 Math
Name: ____________________________ Date: ______________

Problems

Suppose a watershed group samples the fictional Tolliver Creek six times over one winter, pairing specific conductance (in microsiemens per centimeter, written uS/cm) with laboratory chloride (in mg/L).

Specific conductance (uS/cm) Chloride (mg/L)
300 24
400 41
500 53
600 72
700 84
800 101

Technology fits the least-squares line chloride = 0.152 x conductance - 21.3, with a correlation coefficient of 0.999. Use this line for the problems.

  1. Use the regression line to predict the chloride concentration when the specific conductance reads 550 uS/cm. Round to the nearest tenth.
  2. At the sample where conductance was 600 uS/cm, the measured chloride was 72 mg/L. Compute the predicted value and the residual at that point.
  3. Compute the residual at the sample where conductance was 400 uS/cm and measured chloride was 41 mg/L.
  4. Interpret the slope of the line, 0.152, in the context of these two quantities.
  5. Interpret the y-intercept, -21.3, and explain whether it has a sensible physical meaning for this creek.
  6. The correlation coefficient is 0.999. Explain what this value says about how well the line describes the relationship.
  7. The group flags any predicted chloride above 90 mg/L as a concern for salt-sensitive aquatic life. Write and solve for the specific conductance at which the line predicts a chloride of 90 mg/L. Round to the nearest whole number.
  8. Using your residuals and the correlation coefficient, explain whether the group can trust a quick conductance reading to estimate chloride within the range they sampled.

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: 62.3 mg/L.
    Chloride = 0.152 x 550 - 21.3 = 83.6 - 21.3 = 62.3 mg/L.
  2. Answer: 2.1 mg/L.
    Predicted = 0.152 x 600 - 21.3 = 91.2 - 21.3 = 69.9 mg/L. Residual = measured - predicted = 72 - 69.9 = 2.1 mg/L.
  3. Answer: 1.5 mg/L.
    Predicted = 0.152 x 400 - 21.3 = 60.8 - 21.3 = 39.5 mg/L. Residual = 41 - 39.5 = 1.5 mg/L.
  4. Answer: 0.152 mg/L per uS/cm.
    The slope means that each 1 uS/cm increase in specific conductance is associated with about 0.152 mg/L more chloride in the creek.
  5. Answer: not physically meaningful.
    The intercept, -21.3 mg/L, is the predicted chloride at zero conductance. Natural water always has some conductance, and a negative chloride is impossible, so the intercept is an extrapolation artifact outside the sampled range, not a real value.
  6. Answer: 0.999 shows a very strong positive fit.
    A correlation coefficient of 0.999 is very close to 1, so the points lie almost exactly on a rising straight line. Specific conductance and chloride move together very tightly in this data set, so the line describes the relationship extremely well.
  7. Answer: 732 uS/cm.
    Set 0.152x - 21.3 = 90. Then 0.152x = 111.3, so x = 111.3 / 0.152 = 732.2, about 732 uS/cm.
  8. Answer: yes, within the sampled range.
    The residuals in problems 2 and 3 are only about 1.5 to 2.1 mg/L, small compared with the chloride values, and the correlation coefficient of 0.999 is nearly perfect. Together these show the line predicts chloride reliably from conductance across the range sampled, though it should not be trusted far outside 300 to 800 uS/cm.

Differentiation

Support

Provide the regression line already entered so students only substitute values, and work problem 1 as a model. Give a labeled reminder that a residual equals the measured value minus the predicted value.

Extension

Have students enter the six data points into a graphing calculator or spreadsheet, confirm the slope, intercept, and correlation coefficient themselves, then plot the six residuals and describe whether any pattern in the residuals suggests the straight line is missing something.

ELL supports

Pre-teach regression line, predict, residual, and correlation coefficient with a simple scatter plot showing a line and the vertical gap between a point and the line. Allow problems 4, 5, 6, and 8 to be answered with short phrases using provided sentence frames.