Compare whether means of two distributions are the same Which two distributions are you going to compare? I only see one distribution, namely the observed T(t).
Now, as for non-linear models, of course M(t) should be a weather model that is based on some equations of atmospheric physics. But we are not considering the question of how to compute M(t). We are simply asking the question: does the observed temperature T(t) exhibit a growth trend or not, i.e. can we say that, on average, the temperature grows by X degrees per century. This implies a simple linear model, T(t) = a + b*t + U(t) where U(t) is some zero-based random noise.
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Which two distributions are you going to compare? I only see one distribution, namely the observed T(t).
Now, as for non-linear models, of course M(t) should be a weather model that is based on some equations of atmospheric physics. But we are not considering the question of how to compute M(t). We are simply asking the question: does the observed temperature T(t) exhibit a growth trend or not, i.e. can we say that, on average, the temperature grows by X degrees per century. This implies a simple linear model, T(t) = a + b*t + U(t) where U(t) is some zero-based random noise.