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

\left( \mathrm { VO } _ { 2 } \max \right)

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

\mathrm { VO } _ { 2 } \max

in terms of easily obtained quantities. Consider the variables

y = V O _ { 2 } \max ( L / \min )

x _ { 1 } = \text { weight (kg) }

x _ { 2 } = \operatorname { age } ( \mathrm { y } )

x _ { 3 } = \text { time necessary to } \text { walk } 1 \mathrm {~m} \text { ile (min) }

x _ { 4 } = \text { heart } \mathrm { r } \text { ate at the end of the } \text { walk (beats/min) }

Here is one possible model, for male students:

Y = 5.0 + .015 x _ { 1 } - .053 x _ { 2 } - .134 x _ { 3 } - .011 x _ { 4 } + \varepsilon

, and

\sigma = .4

a. Interpret

\beta _ { 1 } \text { and } \beta _ { 3 }

.
b. What is the expected value of

\mathrm { VO } _ { 2 } \max

when weight 75 kg. age is 20 yr, walk time is 15 minutes, and heart rate is 140 b/m?
c. What is the probability that

\mathrm { VO } _ { 2 } \max

will be between 1.00 and 2.60 for a single observation made when the values of the predictors are as stated in part (b)?

The Regression Coefficient β2\beta _ { 2 }β2​ In the Multiple Regression Model

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The Regression Coefficient $\beta _ { 2 }$ In the Multiple Regression Model

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