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A thermometer always reads 0.5 degrees too high…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A thermometer always reads 0.5 degrees too high. The true temperature of a bath is 37.0 degrees. The random scatter of a reading is 0.4 degrees. A student takes 100 readings and reports the mean with its standard error. Simulate this, and say whether the true value lies within two standard errors of the mean. What does the answer teach about averaging?
The worked solution
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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Plan. Each reading is the true value plus 0.5 plus a random error. Compute the mean and the standard error , and see how many standard errors the truth is away from the mean.
Check. The mean is about 37.5 and the standard error is about 0.04. The true value is more than 10 standard errors away, so the report "37.5 plus or minus 0.04" is wrong, even though it looks very precise. The reading is precise and not accurate.
Answer: no, the truth is not within two standard errors (it is more than 10 away). Averaging reduces only random errors. A systematic error, such as a fixed offset, stays, so the thermometer must be calibrated against a known standard.
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