options("scipen" = 9999, "digits" = 4)This chapter covers the basics of parameter estimation, particularly focusing on how to compute unweighted and weighted averages (resulting in the formula for the discrete expected value).
Q1
It’s possible to get errors that don’t quite cancel out the way we want. In the Fahrenheit temperature scale, 98.6 degrees is the normal body temperature and 100.4 degrees is the typical threshold for a fever. Say you are taking care of a child that feels warm and seems sick, but you take repeated readings from the thermometer yourself and get several readings between 97.5 and 98. What could be wrong with the thermometer?
The therometer might be miscalibrated, giving us errors which are systematically biased to be lower than the true values we are tying to measure. So even if the child has a fever, the therometer might not tell us that.
Q2
Given that you feel healthy and have traditionally had a very consistently normal temperature, how could you alter the measurements 100, 99.5, 99.6, and 100.2 to estimate if the child has a fever?
I don’t really think the wording of this question makes any sense, it’s hard to tell what it is actually asking. If we had a consistently normal temperature and weren’t sick, but we got those measurements, we could subtract the bias in the measurement to correct for it. For example, if we subtract 1.5 degrees from all of those measurements, we get temperatures that are all in a normal range, indicating that the thermometer might be miscalibrated.
However, the question asks how we can use this information to measure whether the child has a fever, but in question 1 we were worried about the thermometer giving erroneously low readings. So it’s not really clear what the question wants us to do. Either way, if we have a way to estimate the bias of the thermometer (e.g. by taking temperatures with multiple thermometers or by comparing the current readings with earlier readings when the child was healthy) we could use that bias estimate to correct the measurements we got. However, that would assume that the bias is constant with the temperatures which is not necessarily true, the thermometer might be really bad at reading temperatures above a certain threshold instead. We don’t have nearly enough information in this problem to figure out what is wrong with the child other than they feel warm and seem sick, so we should instead use our best non-thermometer related judgement to decide what to do.