Question
Two sensor teams test the same three water samples. Each sensor reports the true concentration, in parts per million (ppm), plus its own constant additive offset. Measurements are otherwise exact:
| Sample | Sensor A | Sensor B |
|---|---|---|
| 1 | 57 | 52 |
| 2 | 62 | 57 |
| 3 | 54 | 49 |
Can the team infer either sensor's offset or any sample's true concentration from these paired readings alone? Later, sensor A measures a certified reference and reports 47 ppm. The certificate guarantees only that the reference's true concentration lies between 39 and 41 ppm (inclusive), with 40 ppm as its nominal value.
Find the sensor-offset difference, the nominal offsets and the nominal true concentrations of the three samples. Also give the precise remaining concentration ranges implied by the certificate; do not silently treat its nominal 40 ppm as exact.
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Hint 1
Write A's and B's readings as \(x_i+a\) and \(x_i+b\). Look for a change to the unknowns that leaves every shared reading unchanged.
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Solution
Let \(x_1,x_2,x_3\) be the true sample concentrations and \(a,b\) the sensor offsets. The six equations are \(y_{Ai}=x_i+a\) and \(y_{Bi}=x_i+b\). Subtracting any pair gives \(a-b=5\) ppm. They are consistent across all three samples, but do not uniquely determine either offset or the \(x_i\): for any real \(h\), replacing \((x_i,a,b)\) with \((x_i+h,a-h,b-h)\) leaves every measurement unchanged. Thus the six-row linear system in five unknowns has rank four, with one unidentified common-shift direction.
Let \(c\) be the reference's true concentration. Since A reads 47,
\[
a=47-c,\qquad 39\le c\le41
\quad\Rightarrow\quad 6\le a\le8.
\]
Because \(b=a-5\), \(b\in[1,3]\). At the nominal \(c=40\), the calibrated offsets are \(a=7,b=2\) ppm, giving
\[
(x_1,x_2,x_3)=(57,62,54)-7=(50,55,47)\ \text{ppm}.
\]
Respecting the certificate uncertainty instead gives
\[
x_1\in[49,51],\quad x_2\in[54,56],\quad x_3\in[46,48]\ \text{ppm}.
\]
The differences among sample concentrations (5, −3 and −8 ppm, depending on order) are exact: each true level shifts by the same unknown amount. Thus the anchor identifies absolute levels only up to its certified ±1 ppm tolerance. Treating repeated paired samples as extra calibration references would falsely remove this remaining uncertainty.
Assumptions
Each unknown sample concentration is common to both readings, and each instrument has one constant offset for all readings.
Readings are otherwise exact; the only range uncertainty is the certified interval 39–41 ppm.
The reference is measured only by sensor A.
Final Answer
Paired readings identify \(a-b=5\) ppm, but no absolute offsets without the reference. Nominally \((a,b)=(7,2)\) ppm and true samples \((50,55,47)\) ppm. Exact identified ranges: \(a\in[6,8]\), \(b\in[1,3]\), samples \([49,51], [54,56], [46,48]\) ppm.