Question b Capture Pro Fell running is a sport in which competitors race each other on foot over hills and mountains. Organisers of such events want to understand how the time taken by elite competitors depends on the route measured in terms of horizontal distance run and the height climbed by the competitors during the event. (Assume that the courses are well marked and competitors must keep to the marked routes). Data have therefore been collected on the times taken by the top ten finishers in 8 different events. The information recorded for each runner in each event was: distance - horizontal distance in miles . height - vertical distance in '000 feet time - runner's finishing time in minutes. The basic descriptive statistics for these variables are given below. Descriptive Statistics Std. N Minimum Maximum Mean Deviation DISTANCE 80 6.00 12.00 9.3750 2. 1310 HEIGHT 80 1.20 5.30 2.6250 1.3777 TIME 80 42.37 143.85 82.9404 28. 1408 Valid N (listwise 80 Four simple rules have been proposed to convert the height climbed into an equivalent horizontal distance: Rule1: just measure horizontal distance; Rule2: add 1.1 miles to the horizontal distance for each 1000 feet climbed; Rule3: add 1.2 miles to the horizontal distance for each 1000 feet climbed; Rule4: add 1.3 miles to the horizontal distance for each 1000 feet climbed. SPSS has then been used to regress competitors' times against the four 'equivalent' horizontal distances produced by each of these rules (referred to as variables RULE1, RULE2, RULE3, RULE4). The statistical outputs plus some scatterplots (with regression lines marked) are shown below. (a) According to the statistical outputs, which of these four sets of results show evidence of a significant linear relationship between time and the explanatory rule? Justify your answer. (3 marks) (b) Compare the four rules on the basis of the statistical outputs provided. Which of the four rules do you think is best on statistical grounds? Justify your answer. (3 marks) (c) Explain the coefficients of your chosen model as if to an organiser of the event. Comment on any apparent peculiarities in your model. (4 marks)Apowers (d) Referring to the scatterplots comment carefully on whether or not the usual regression assumptions seem to hold for your preferred model. Explain why each graph shows the data in eight vertical stripes. (5 marks) (e) The event organisers are concerned that none of the four proposed rules make any direct allowance for fatigue, whereas as it is well known that as races get longer runners typically run more slowly. Explain the extent (if any) to which your preferred model incorporates effects of fatigue, and suggest a type of rule that could be developed to better reflect the effects of fatigue. (10 marks) Statistical Outputs Rule1 Model Summary Adjusted Std. Error of Mode R R Square R Square the Estimate 1 8678 751 748 14.1284 a. Predictors: (Constant), RULE1 ANOVA Sum of Model Squares Mean Square F Sig. Regression 46990.88 1 46990.885 235.413 0003 Residual 15569.64 78 199.611 Total 62560.53 79 a. Predictors: (Constant), RULE1 b. Dependent Variable: TIME Coefficients Standardi zed Unstandardized Coefficie Coefficients nts Model B Std. Error Beta Sig. Constant) -24.355 7.169 3.397 00 1 RULE1 11.445 746 867 15.343 000 a. Dependent Variable: TIMEowersoft been Capture Pro Model Summary Adjusted Std. Error of Model R R Square R Square the Estimate 9738 946 945 6.5922 a. Predictors: (Constant), RULE2 ANOVA Sum of Model Squares if Mean Square F Sig Regression 59170.83 59170.829 136 1.573 000 Residual 3389.700 78 43.458 Total 62560.53 79 a. Predictors: (Constant), RULE2 b. Dependent Variable: TIME Coefficients Standardi zed Unstandardized Coefficie Coefficients nts Model B Std. Error Beta Sig. (Constant) -18.191 2.838 -6.410 000 RULE2 8.192 .222 973 36.900 000 a. Dependent Variable: TIME Rule3 Model Summary Adjusted Std. Error of Model R R Square R Square the Estimate 9758 950 950 6.3120 a. Predictors: (Constant), RULE3 ANOVA Sum of Model Squares Mean Square F Sig Regression 59452.91 59452.909 1492.244 0003 Residual 3107.621 78 39.841 Total 62560.53 79 a. Predictors: (Constant), RULE3 b. Dependent Variable: TIME Coefficients Standardi zed Unstandardized Coefficie Coefficients nts Model B Std. Error Beta Sig. Constant) -16.976 2.681 6.332 000 RULE3 7.920 205 975 38.630 000 a. Dependent Variable: TIMERule4 pture Pro Model Summary Adjusted Std. Error of Model R R Square R Square the Estimate .977" 954 953 6.0819 a. Predictors: (Constant), RULE4 ANOVA Sum of Model Squares Mean Square F Sig. Regression 59675.36 1 59675.357 1613.310 000 Residual 2885.173 78 36.989 Total 62560.53 79 a. Predictors: (Constant), RULE4 b. Dependent Variable: TIME Coefficients Standardi Zed Unstandardized Coefficie Coefficients nts Mode B Std. Error Beta Sig. [Constant) -15.770 2.550 6.185 000 RULE4 7.661 191 977 40.166 000 a. Dependent Variable: TIME Scatterplots Linesar Regression 1:25.00 time = -24.36 + 11.44 * rykaf R-Square = 0.75 time 75.00- 50100- 10 00 12.00 rule1een Capture Linear Regression 12500- time =-18.19 + 8.19 rule A-Square = 0.53 time 90 00- 10.0 12 60 14 0 18 00 rule2 Linear Regression time 50.00- 10.00 14/60 18.00 rule3 Linear Regression 18400- time =-15.27 + 7.60 * rule A-3quard = 0.56 time 50 00 18.08 12 56 1750 rule4