Stream Sensors and a Missing Week
A class monitors a local stream for a term project. A probe logs water temperature every 15 minutes. During week 6, the logger's battery failed, so no readings exist for seven days. The students also have weekly hand measurements of temperature taken at 9 a.m. every Monday, plus daily air-temperature records from a nearby weather station. The stream temperature normally tracks air temperature with a lag of a few hours and is damped, staying cooler than air in summer afternoons. The class must report a weekly mean stream temperature for week 6 to compare against weeks 1–10. One student suggests simply using the Monday hand reading as the week-6 value. Another suggests leaving week 6 blank.
1. Evaluate the proposal to use the single Monday hand reading as the week-6 mean.
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A single 9 a.m. reading is a biased and imprecise estimate of a weekly mean. Stream temperature has a strong daily cycle; a morning value is typically near the daily minimum, while afternoon values are higher. Using only 9 a.m. data would systematically underestimate the weekly mean in summer.
It is also one observation standing in for 672 logged readings, so sampling error is large. The estimate is not comparable to other weeks, which are means of 15-minute data. Substituting it would create a spurious dip or shift in the week-6 comparison and could distort any trend.
2. Design a better way to estimate the week-6 mean using the air-temperature record, and state the key assumption.
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Use the other nine weeks to build a regression model predicting 15-minute stream temperature from air temperature (including a lagged term to capture the few-hour delay). Fit it on weeks with complete data, then apply it to week 6's air-temperature series to predict stream values and average them.
Key assumption: the relationship between air and stream temperature in week 6 is the same as in the calibration weeks. This fails if a storm, groundwater pulse, or shading change altered the stream. Validate by checking model fit on held-out weeks and report a prediction interval, not just a point estimate.
3. What should the report say about uncertainty for week 6, and how should the comparison across weeks be presented?
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The week-6 value should be reported as an estimate with a range, e.g., a predicted mean plus a prediction interval, and clearly flagged as modeled rather than measured. The report should state the battery failure as a data-quality issue.
For the 10-week comparison, present measured weeks as means with variability (standard deviation or standard error) and week 6 as a modeled point with its interval, ideally shown as a distinct marker. Conclusions about trends should note that week 6 carries more uncertainty and should not drive the interpretation alone.