Statistics 101-106, Fall 1998

Pollard homework sheet 10

Due: Thursday 19 November

For each of 12 children, the columns denote the height of the child (inches), the weight of the child (pounds) and the distance to the pulmonary artery (centimeters).

 
height weight distance
42.8 40.0 37.0
63.5 93.5 49.5
37.5 35.5 34.5
39.5 30.0 36.0
45.5 52.0 43.0
38.5 17.0 28.0
43.0 38.5 37.0
22.5  8.5 20.0
37.0 33.0 33.5
23.5  9.5 30.5
33.0 21.0 38.5
58.0 79.0 47.0

Write yi for the distance, hi for the height, and wi for the weight of the ith child.

The Minitab output from the regression of distance on height and weight contains the following:

The regression equation is
distance = 21.0 + 0.196 height + 0.191 weight

S = 3.943       R-Sq = 80.5%     R-Sq(adj) = 76.2%

Source            DF          SS          MS         F        P
Regression         2      578.82      289.41     18.62    0.001
Residual Error     9      139.91       15.55
Total             11      718.73

Source       DF      Seq SS
height        1      558.06
weight        1       20.75

1. Regress distance on height, getting a fitted line yi = a0 +a1*hi, with vector of residuals f.
2. Regress weight on height, getting a fitted line wi = c0 + c1* hi, with vector of residuals g.
3. Regress f on g, getting a fitted line fi = d0 + d1*gi, with vector of residuals e.
4. Explain why the regression line in (iii) must pass through the origin. That is, explain why d0 must equal 0.
5. Check that e is the vector of residuals obtained from regressing y on h and w.
6. Show that the equation for the fitted value,

   21.0 + 0.196 height + 0.191 weight,
   

   can be obtained from the coefficients a0, a1, c0, c1, and d1. Hint: Eliminate f and g from the equations in (i), (ii), and (iii).
7. Explain how the R-squared value 80.5% could be calculated from the R-squared values from steps (i) and (iii). Hint: Think of what 1-R-squared tells you about ratios of sums of squares.
8. Explain how the final decomposition

   Source       DF      Seq SS
   height        1      558.06
   weight        1       20.75
   

   is related to steps (i), (ii), (iii).