Question 3 (25 Marks). Sixteen male well-trained middle and long distance runners performed a 3 km time trial and a numb

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Question 3 25 Marks Sixteen Male Well Trained Middle And Long Distance Runners Performed A 3 Km Time Trial And A Numb 1 (106.88 KiB) Viewed 40 times

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Question 3 (25 Marks). 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? [2 marks] b) What is the observed correlation between VO2Max and 3km finishing time? [1 mark) c) Based on the p-values presented, explain why there is evidence that VO2Max is a significant predictor of 3km finishing? [6 marks] d) Provide an interpretation of the R-sq statistic in terms of how useful is VO2Max as a predictor of 3km finishing time. [2 marks) e) Provide an interpretation of the residual standard error (highlighted in bold) in terms of using this model to predict 3km finishing time. [2 marks] 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. [4 marks) g) What are the assumptions underlying the model presented and do they look justified based on the residual plots provided? [8 marks]

Scatterplot with Lowess Smoother and Line of Best Fit (uncertainty band is for the smoother) 11- 10 3km Running Time (mins) 16 18 22 20 VO2Max (ml/kg. min)

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(>1t1) (Intercept) 15.69623 0.97466 16.104 1.98e-10 *** VO2Max -0.30154 0.04688 -6.432 1.57e-05 *** Signif. codes: ***** 0.001 "**' 0.01 *** 0.05 '.' 0.1'' 1 Residual standard error: 0.3874 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") upr fit lwr 1 10.268513 9.370485 11.16654 29.665433 8.806241 10.52463 39.363894 8.506911 10.22088

Scatterplot with line of best fit and Prediction Intervals 11 3km Running Time (mins) 8- - - 16 18 22 20 VO2Max (ml/kg. min)

Residuals vs Fitted Normal Q-Q 1.0 016 100 90 0.5 09 O Residuals -1 0 1 2 o Standardized residuals 00 o 00 OOOO..0.0.0 100 O o O -05 130 7- 013 9 9.0 9.5 10.0 10.5 -2 -1 0 1 2 Fitted values Im(Running_Time_3km - VO2Max) Theoretical Quantiles Im(Running_Time_3km - VO2Max) Scale Location 016 Residuals vs Leverage 1.5 130 016 09 o 1 2 0 o 105 10 o O o VIStandardized residuals Standardized residuals O o O OPO 05 0 O -2 -1 0 O O O 1 00 Cook's distance 130 9.0 95 100 10.5 0.00 0.05 0.10 0.15 0.20 0.250.300.35 Fitted values Im(Running Time_3km - VO2Max) Leverage Im(Running Time 3km - VO2Max)