Linear Regression Worksheet 1


Does a person's muscle mass decrease with age? To explore this relationship in women, a nutritionist randomly selected four women from each 10-year age group, beginning with age 40 and ending with age 79.

Obsn  mus_mass   age
   1        82    71
   2        91    64
   3       100    43
   4        68    67
   5        87    56
   6        73    73
   7        78    68
   8        80    56
   9        65    76
  10        84    65
  11       116    45
  12        76    58
  13        97    45
  14       100    53
  15       105    49
  16        77    78

The least squares regression line is: Y-hat = 148.0 - 1.023 x. (Y-hat stands for a Y with a caret (^) over it; the predicted value).


  1. The first observation has age = 71.0 and muscle mass = 82. Identify this value on the plot.
  2. Carefully plot the line. Use x = 40 and x = 80 to do so.
  3. From your plot alone, predict the muscle mass of a 60 year old.
  4. Use the equation to check your answer to part 3.
  5. Complete the table below. For each observation the table lists the age and muscle mass. You should fill in the predicted value and residual. (Some have been filled in for you.)
  6. Sum the residuals. What do you get?
  7. Sort the residuals.
  8. Report an 80% prediction interval for the muscle mass of a 60 year old. Plot the interval.
  9. Report an 80% prediction interval for the muscle mass of a 43 year old. Plot the interval.
  10. Complete your plot by drawing 80% prediction bands.
  11. What %-age of the data fall within these prediction bands? (Count an observation falling right on a band as being 1/2-in and 1/2 out.)
Age Muscle Mass Predicted Y Residual
71 82 75.4 6.6
64 91 82.5 8.5
43 100 104.0 -4.0
67 68 79.5 -11.5
56 87 90.7 -3.7
73 73 73.3 -0.3
68 78    
56 80    
76 65    
65 84    
45 116    
58 76    
45 97 102.0 -5.0
53 100 93.8 6.2
49 105 97.9 7.1
78 77 68.2 8.8

ANSWERS

1. 86.6
6. Sum = 0. (It works out to 0.1 due to accumulated rounding errors in the residuals.)
8. (75.1, 95.4). (The "official method"-covered in Math 258-yields (75.0, 98.2).)
9. (92.5, 112.8)