Sixteen male well-trained middle and long distance runners performed a 3 km time trial and a number of running tests in
Posted: Mon May 02, 2022 6:46 am
Sixteen male well-trained middle and long distance runners
performed a 3 km time trial and a number of running tests in the
laboratory where their VO2Max (i.e. the maximum or optimum rate at
which the heart, lungs, and muscles can effectively use oxygen
during exercise) was measured as a way of measuring their
individual aerobic capacity. All the laboratory testing took place
on a motorised treadmill while distance running performance was
determined by 3 km time trials on an indoor 200m track. The aim of
the study was to investigate whether there is sufficient evidence
of a dependency of 3 km running time on VO2Max in the population of
male runners of interest in order to use their aerobic capacity to
predict their 3km running time. A scatterplot (with lowess smoother
and line of best fit) is provided, as is output from a regression
analysis carried out on these data.
a) What are the slope and intercept of the least-squares
line?
b) What is the observed correlation between VO2Max and 3km
finishing time?
c) Based on the p-values presented, explain why there is
evidence that VO2Max is a significant predictor of 3km
finishing?
d) Provide an interpretation of the R-sq statistic in terms of
how useful is VO2Max as a predictor of 3km finishing time.
e) Provide an interpretation of the residual standard error
(highlighted in bold) in terms of using this model to predict 3km
finishing time.
f) A particular athlete recorded a VO2Max of 20 in the lab prior
to a 3km event. Use the output below to provide a range of
predicted values for his likely finishing time.
g) What are the assumptions underlying the model presented and
do they look justified based on the residual plots provided?
Scatterplot with Lowess Smoother and Line of Best Fit (uncertainty band is for the smoother) 11- 3km Running Time (mins) 8- 1 - 16 18 20 VO2Max (ml/kg. min) 22
1m (formula - Running_Time_3km - VO2Max, data = running.df) Residuals: Min 1Q Median 3Q Max -0.65128 -0.29101 0.03749 0.15649 0.85426 Coefficients: Estimate Std. Error t value Pr(>t|) (Intercept) 15.69623 0.97466 16.104 1.98e-10 *** VO2Max -0.30154 0.84688 -6.432 1.57e-05 *** Signif. codes: @"***0.001 0.01 0.05'' 0.1''1 Residual standard error: 6.3874 on 14 degrees of freedom Multiple R-squared: 0.7472, Adjusted R-squared: 0.7291 F-statistic: 41.37 on 1 and 14 DF, p-value: 1.569e-05 running_new <- data.frame(VO2Max = c(18, 20, 21)) predict(running.model, newdata = running_new, interval = "confidence") fit lwr upr 1 10.268513 9.927684 10.609342 2 9.665433 9.446523 9.884344 3 9.363894 9.153821 9.573966 predict(running.model, newdata = running_new, interval = "prediction") fit lwr upr 1 10.268513 9.370485 11.16654 2 9.665433 8.806241 10.52463 3 9.363894 8.506911 19.22088
Scatterplot with line of best fit and Prediction Intervals 11 - 10- 3km Running Time (mins) 8- - 16 18 22 20 VO2Max (ml/kg. min)
- Residuals vs Fitted Normal Q-Q 10 016 180 2 H 90 05 1 I Residuals Standardized residuals 00 0 OOO -1 0 0 -0.5 130 z 013 T 10.5 9.0 9.5 100 2. -1 0 2 Fitted values Im(Running_Time_3km - VO2Max) Theoretical Quantiles Im(Running_Time_3km - VO2Max) Scale-Location Residuals vs Leverage S 016 130 016 2 00 0.5 10 o IStandardized residuals Standardized residuals U O SO O 2 -1 0.0 Cook's distance 130 9.0 9.5 10.0 10.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Fitted values Im(Running_Time_3km - VO2Max) Leverage Im(Running_Time_3km - VO2Max)
performed a 3 km time trial and a number of running tests in the
laboratory where their VO2Max (i.e. the maximum or optimum rate at
which the heart, lungs, and muscles can effectively use oxygen
during exercise) was measured as a way of measuring their
individual aerobic capacity. All the laboratory testing took place
on a motorised treadmill while distance running performance was
determined by 3 km time trials on an indoor 200m track. The aim of
the study was to investigate whether there is sufficient evidence
of a dependency of 3 km running time on VO2Max in the population of
male runners of interest in order to use their aerobic capacity to
predict their 3km running time. A scatterplot (with lowess smoother
and line of best fit) is provided, as is output from a regression
analysis carried out on these data.
a) What are the slope and intercept of the least-squares
line?
b) What is the observed correlation between VO2Max and 3km
finishing time?
c) Based on the p-values presented, explain why there is
evidence that VO2Max is a significant predictor of 3km
finishing?
d) Provide an interpretation of the R-sq statistic in terms of
how useful is VO2Max as a predictor of 3km finishing time.
e) Provide an interpretation of the residual standard error
(highlighted in bold) in terms of using this model to predict 3km
finishing time.
f) A particular athlete recorded a VO2Max of 20 in the lab prior
to a 3km event. Use the output below to provide a range of
predicted values for his likely finishing time.
g) What are the assumptions underlying the model presented and
do they look justified based on the residual plots provided?
- Sixteen Male Well Trained Middle And Long Distance Runners Performed A 3 Km Time Trial And A Number Of Running Tests In 1 (43.66 KiB) Viewed 46 times
- Sixteen Male Well Trained Middle And Long Distance Runners Performed A 3 Km Time Trial And A Number Of Running Tests In 2 (54.76 KiB) Viewed 46 times
1m (formula - Running_Time_3km - VO2Max, data = running.df) Residuals: Min 1Q Median 3Q Max -0.65128 -0.29101 0.03749 0.15649 0.85426 Coefficients: Estimate Std. Error t value Pr(>t|) (Intercept) 15.69623 0.97466 16.104 1.98e-10 *** VO2Max -0.30154 0.84688 -6.432 1.57e-05 *** Signif. codes: @"***0.001 0.01 0.05'' 0.1''1 Residual standard error: 6.3874 on 14 degrees of freedom Multiple R-squared: 0.7472, Adjusted R-squared: 0.7291 F-statistic: 41.37 on 1 and 14 DF, p-value: 1.569e-05 running_new <- data.frame(VO2Max = c(18, 20, 21)) predict(running.model, newdata = running_new, interval = "confidence") fit lwr upr 1 10.268513 9.927684 10.609342 2 9.665433 9.446523 9.884344 3 9.363894 9.153821 9.573966 predict(running.model, newdata = running_new, interval = "prediction") fit lwr upr 1 10.268513 9.370485 11.16654 2 9.665433 8.806241 10.52463 3 9.363894 8.506911 19.22088
Scatterplot with line of best fit and Prediction Intervals 11 - 10- 3km Running Time (mins) 8- - 16 18 22 20 VO2Max (ml/kg. min)
- Residuals vs Fitted Normal Q-Q 10 016 180 2 H 90 05 1 I Residuals Standardized residuals 00 0 OOO -1 0 0 -0.5 130 z 013 T 10.5 9.0 9.5 100 2. -1 0 2 Fitted values Im(Running_Time_3km - VO2Max) Theoretical Quantiles Im(Running_Time_3km - VO2Max) Scale-Location Residuals vs Leverage S 016 130 016 2 00 0.5 10 o IStandardized residuals Standardized residuals U O SO O 2 -1 0.0 Cook's distance 130 9.0 9.5 10.0 10.5 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Fitted values Im(Running_Time_3km - VO2Max) Leverage Im(Running_Time_3km - VO2Max)