Example 3. Simple Inverse Linear Regression: Steam Data

The data shows observations taken at intervals from a steam plant which is part of a large industry. This data is in fact a portion of a larger data set used to fit a multiple regression model. The response variable and predictor variable are:
Y: Monthly use of steam (pounds)
X: Average atmospheric pressure (degrees F)

Source: Draper, N.R. and Smith, H. (1998). Applied Regression Analysis. John Wiley & Sons: New York.

Table 3: Portion of the Steam Data

X      Y
35.3  10.98 
29.7  11.13 
30.8  12.51 
58.8   8.40 
61.4   9.27 
71.3   8.73 
74.4   6.36 
76.7   8.50 
70.7   7.82 
57.5   9.14 
46.4   8.24 
28.9  12.19 
28.1  11.88 
39.1   9.57 
46.8  10.94 
48.5   9.58 
59.3  10.09 
70.0   8.11 
70.0   6.83 
74.5   8.88 
72.1   7.68 
58.1   8.47 
44.6   8.86 
33.4  10.36 
28.6  11.08

Questions:

1. Suppose we specify a true mean value of Y₀=10. How can we construct a point estimate and an interval estimate of X₀ corresponding to Y₀?
   
   
2. How would our result in (1) change if Y₀ is not the true mean value, but rather a single observation?

Keywords: Inverse regression, inverse prediction limits