MMP.13:5.2 - A common calibration error survives averaging

Four readings in fixed units are 9, 10, 10 and 11, giving mean 10. Initially suppose

Y_i=mu+epsilon_i; epsilon_i independently Normal(0,1).

The noise variance 1 is supplied, not estimated from these four values. The sample mean is an estimator of mu with variance 1/4. An exact normal 95% confidence procedure uses mean(Y) +/- c/2, where c is the 0.975 standard-normal quantile, approximately 1.96. The realized interval is approximately [9.020,10.980].

A prediction interval for one independent future reading uses the error Y_new-mean(Y), whose variance is 1+1/4. Its realized 95% interval is approximately [7.809,12.191]. Uncertainty about the mean and variation of the future reading require different intervals even before any model revision.

Now revise the calibration account:

Y_i=mu+B+epsilon_i; B~Normal(0,1).

B is independent of the individual errors, shared by all readings in one setup, and drawn afresh across the repeated setups used to define the coverage claim. Then

Var(mean(Y))=1+1/4.

The confidence interval for mu widens to [7.809,12.191]. Treating B as a fresh independent error on each row would instead give variance 2/4 and understate the uncertainty. More readings in this same setup reduce the individual-noise term but leave the calibration term.

Prediction also depends on what stays shared. A new reading in the same setup has the same B, which cancels in Y_new-mean(Y); the prediction-error variance remains 1+1/4. A reading in a new setup with independent B_new has variance 1+1+1+1/4=3.25 for that error, giving approximately [6.467,13.533].

These are coverage statements over the stated repeated-observation law. If B is only an unknown fixed offset with no bound or probability law, these normal intervals for mu do not follow. MMP.12 then retains the unresolved separation of mu and B; assigning B a distribution is an additional modeling contribution.