10.10 Problems

1. Fit a straight line model to the data given in Table 10.3, where d _i is the average of n _i data values measured at x _i. The probability of the individual d _i measurements is normal with ? = 4.0, regardless of the x _i value.

   1. Give the slope and intercept of the best-fit line together with their errors.

   2. Plot the best-fit straight line together with the data values and their errors.

   3. Give the parameter covariance matrix.

   4. Repeat (a) and (c) but this time use the average x-coordinate as the origin. Comment on the differences between the covariance matrices in (c) and (d).

      Table 10.3: Data table

      x i	d i	n i
      10	0.387	14
      20	5.045	3
      30	7.299	25
      40	6.870	2
      50	16.659	3
      60	13.951	22
      70	16.781	5
      80	20.323	2

2. Compute and plot the ellipse that defines the 68.3% and 95.4% joint credible region for the slope and intercept, for the data given in Table 10.3. The shape of this ellipse depends on the x-coordinate origin used in the fit (see Figure 10.5). Use the average x-coordinate as the origin. See the section of the Mathematica tutorial entitled Joint Credible Region Contouring.

3. Table 10.4 gives measurements of ozone partial pressure, y, in millibars in each of 15 atmospheric layers where each layer, x, is approximately 2 km in height. The layers have been scaled for convenience from ?