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In: Statistics and Probability

Cardiorespiratory fitness is widely recognized as a major component of overall physical well-being. Direct measurement of...

Cardiorespiratory fitness is widely recognized as a major component of overall physical well-being. Direct measurement of maximal oxygen uptake (VO2max) is the single best measure of such fitness, but direct measurement is time-consuming and expensive. It is therefore desirable to have a prediction equation for VO2max in terms of easily obtained quantities. Consider the variables

y = VO2max (L/min)
x1 = weight (kg)
x2 = age (yr)
x3 = time necessary to walk 1 mile (min)
x4 = heart rate at the end of the walk (beats/min)

Consider the model of the entire population.

Y = β0 + β1x1 +β2x2 + β3x3 + β4x4 + ε

:
Here is one possible model for male students:

Y = 5.0 + 0.019x1 − 0.05x2 − 0.12x3 − 0.01x4 + ε


σ = 0.4

(a) Interpret β1.

Holding all other variables constant, a 1 kg increase in weight will result in a 0.019 L/min decrease in VO2max.Holding all other variables constant, a 1 kg increase in weight will result in a 0.019 L/min increase in VO2max.    Holding all other variables constant, a 0.019 kg increase in weight will result in a 1 L/min increase in VO2max.Holding all other variables constant, a 0.019 kg increase in weight will result in a 1 L/min decrease in VO2max.


Interpret β3.

Holding all other variables constant, a 0.12 min increase in walk time will result in a 1 L/min decrease in VO2max.Holding all other variables constant, a 0.12 min increase in walk time will result in a 1 L/min increase in VO2max.    Holding all other variables constant, a 1 min increase in walk time will result in a 0.12 L/min increase in VO2max.Holding all other variables constant, a 1 min increase in walk time will result in a 0.12 L/min decrease in VO2max.


(b) What is the expected value of VO2max, μy, when weight is 79 kg, age is 25 yr, walk time is 14 min, and heart rate is 140 b/m?
L/min

(c) What is the probability VO2max is between 1.531 and 2.811, when weight is 79 kg, age is 25 yr, walk time is 14 min, and heart rate is 140 b/m?  Hint: The response varies normally with a mean of μy and a standard deviation of σ.

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