Question

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, μ_y, 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 μ_y and a standard deviation of σ.

Answer 1

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