Reference no: EM132856967
This is a subset of a data set originally published by Atkinson (1986) and republished by Hand et al's A Handbook of Small Data Sets. The original data set gives record times in 1984 for 35 hill races (data from Hand et al's A Handbook of Small Data Sets). Three quantitative variables are recorded: Distance (in miles), HeightGain (in feet; this is a measure of total vertical climb over the course of the race), and Time (in minutes; seconds have been converted to decimal minutes). The subset you will be working with has had five observations deleted from the original data set.
(a) Fit two multiple regression models for predicting Time from the two predictor variables, Distance and HeightGain, that differ only in the order the predictor variables enter the model. Obtain key results with the summary() and anova() commands. What conclusions can you make as a result of these analyses?(You do not need to formally state the null and alternative hypotheses for each test as long as you are clear in your conclusions what you have tested. Be sure to include the appropriate test statistics and their associated p-values.)
(b) The results in (a) are distorted by the presence of two extreme outliers. Obtain a box plot of the residuals from one of your models fit in (a). Identify the races that produced the extreme outliers. Why are these outliers so extreme?
(c) In (b), what are the values of the "outer fences" that would classify an outlier as an extreme outlier? Do you think it is a coincidence that both extreme outliers are for positive residuals? Briefly explain your reasoning.
(d) Repeat the analyses in (a) after removing the two extreme outliers identified in (b). How does a boxplot of the residuals look now? What conclusions can you make now as a result of these analyses?
(e) Summarize the key differences in the conclusions you reached in (a) and (d). Include a comment on the effect of including vs excluding the two extreme outliers on the MSE and the coefficient of determination.
(f) It makes sense that longer races will take longer to run, and that the amount of vertical climb over the course of the race might influence the time it takes to finish race. But what are the relative effects of the two predictor variables on the pace the race is run, where Pace = Time/Distance? Repeat your analyses in part (d) but with Pace as the response variable. Summarize your conclusions as concisely yet completely as you can.
(g) According to the website https://www.bennevisrace.co.uk/, the record time for the Ben Nevis race, which was set in 1984 by the Scottish hill racing legend KennyStuart, still stands. This record is included in the data set we have been analysing (but there is a one second discrepancy between our data set and the website time). What is the predicted record Time and Pace for the Ben Nevis race, based on the multiple regression analyses you have run (extreme outliers excluded)?
(h) What additional factors (other than Distance and HeightGain) could contribute to variability in the Time and Pace response variables? You should be able to come up with a few good ones, give us exactly three factors as your solution to this question. Factors which can be measured readily are worth more than those that would be very hard to measure.
(mikedewin)
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