Air pressure as a function of elevation, p. 6

    Residuals from the regression of log(mmHg) on elevation: 
    ------
           [1,]          [2,]          [3,]          [4,]          [5,] 
    -7.293875e-02 -6.711840e-02 -6.157151e-02 -5.642369e-02 -5.109899e-02 
           [6,]          [7,]          [8,]          [9,]         [10,] 
    -4.618506e-02 -4.120840e-02 -3.665898e-02 -3.230053e-02 -2.800848e-02 
           [11,]         [12,]         [13,]         [14,]         [15,] 
    -2.391216e-02 -1.988236e-02 -1.564377e-02 -1.187340e-02 -7.883610e-03 
           [16,]         [17,]         [18,]         [19,]         [20,] 
    -3.799925e-03 -5.127325e-05  3.798401e-03  7.151902e-03  1.107411e-02 
           [21,]         [22,]         [23,]         [24,]         [25,] 
     1.465524e-02  2.172690e-02  2.837088e-02  3.441591e-02  4.040117e-02 
           [26,]         [27,]         [28,]         [29,]         [30,] 
     4.618832e-02  7.084317e-02  8.809542e-02  9.712198e-02  9.698046e-02 
           [31,]         [32,]         [33,]         [34,]         [35,] 
     8.726908e-02  7.059163e-02  5.305062e-02  3.646433e-02  1.968732e-02 
           [36,]         [37,]         [38,]         [39,]         [40,] 
     1.917667e-03 -2.885418e-02 -5.741835e-02 -7.731274e-02 -8.965992e-02 
     -------

     These residuals start out negative and increasing (the 17th
     is just below 0), then they become positive and still increasing
     up to the 29th, then they decrease the rest of the way, becoming
     negative again at the 37th. So there is a strong pattern with
     only one turning point, and the model, despite the high R^2
     and small residuals, can't have i.i.d. normal errors, and none
     of the three models we've considered do for the given data.

     The residuals in the logs correspond to "relative" or percentage
     errors in the actual pressures. From 50,000 feet on up, the table 
     has just three significant figures in the mm Hg pressures (from 70,000 
     feet on up, only two in the pounds per square inch). The residuals
     in the logs are still bigger than the rounding errors in the mm Hg. 
     For the highest elevation, 100,000 feet, the actual error in p = 8.36 
     from the current model is about 0.717. Here the quadratic model has a 
     disastrously large residual of -73.6. The magnitudes of the pressures
     are varying by a factor of more than 100, from 903.7 down to 8.36, 
     so relative errors, namely errors in logs, may make good sense.

     For the highest pressures in the table, at -5,000 feet (about a 
     mile below the earth's surface --- Googling "deepest mine" indicates 
     there are some deeper mines) the error in log(p) of about -0.073 in a 
     pressure of p = 903.7 mm Hg gives an error in p of about 63.6, 
     comparable to the errors in the quadratic regression.