Oxygen intake and heart rate

Use this page to explain that the regression model with β0, β1 and σ replaced by estimates provides an estimate of the response distribution at any new x-value.

The grey band represents the fitted regression line ± twice the estimated standard deviation of the model error. Use the slider to show the estimated normal distribution of oxygen intake at any heart rate.

The mean of this estimated distribution (i.e. the prediction from the least squares line) provides a point prediction.


In athletes, oxygen intake (VO2) and heart rate both increase when exercise is increased. Sports scientists are extremely interested in monitoring VO2, but it is a much harder quantity to measure than heart rate, so an experiment was conducted to examine the relationship between heart rate and VO2 for a single individual. Both variables were recorded for the individual at various work loads.

Oxygen intake Heart rate   Oxygen intake Heart rate
0.473
0.753
0.929
0.939
0.832
0.983
1.049
1.178
1.176
1.292
94
96
95
95
94
95
94
104
104
106
 
1.403
1.499
1.529
1.599
1.749
1.746
1.897
2.040
2.231
 
108
110
113
113
118
115
121
127
131