Gene Gallagher wrote:
> Imagine this as an MCAS question:
>
> A scientist has two thermometers, one Fahrenheit and the other Celsius.
> When the temperature is less than 8.4 degrees, the scientist
> uses the Celsius scale. When the temperature is more than
> 8.4 degrees, he uses the Fahrenheit scale. In 1998, he took 40
> measurements of the temperature. In the two following years, he
> took 40 more measurements of the temperature each year.
> The mean temperature was one degree less in 1998 than the mean
> of the 1999 and 2000 temperatures. Should the scientist:
> a) Conclude that the temperature is getting colder
> b) Conclude that the temperature is about the same
> c) Conclude that the temperature is getting warmer
> d) Not conclude anything since it is meaningless to average
> degrees Farhenheit and degrees Celsius in this way.
Interesting. If it's 8.4 degrees Fahrenheit at which he
switches, no conclusion could be drawn for *any* sample size because the
recorded value would not be a monotonic function of actual temperature.
; if the change takes place at 8.4C the sample size might be
insufficient, but as the scale is monotone (the MCAS score is) valid
conclusions could be drawn from a large enough sample (the ubiquitous
central limit theorem!) - with the understanding that they were not in
general conclusions about the arithmetic mean temperature, but about
another measire of location.
However, we have to understand that in the absence of a well-defined
ONE-parameter family of alternatives (eg, shift, scale, etc) or a class
of distributions such as the symmetric distributions for which
"location" is hard to define in any other way, the assumption that
"score increasing" must mean "arithmetic mean of scores increasing" is
arbitrary.
-Robert Dawson
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