Sample exercise International Bachelor Econometrics and Operations Research - BSc
The brochure on the International Bachelor in Econometrics and Operations Research contains a sample assignment that you can use to test your knowledge of this programme and see which kinds of assignments a student is likely to get.
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Answer to sample exercise of the International Bachelor in Econometrics and Operations Research
The brochure International Bachelor Econometrics and Operations Research contains an example exercise. With this exercise, you can get an idea of the type of questions that students face in their study.
The question is: Are you able to predict the next world record?
1. Make a model
It is helpful to make a graphical display of the data. The next two diagrams show the record time (on the vertical axis) against the calendar year (on the horizontal axis), for men on the left and for women on the right.
The diagram for men indicates that the world record decreased more quickly in the period 1990-2010 than in the period 1970-1990.
The opposite is the case for women, as the world record improved rather steadily until 1988 but it did not yet improve since then. In the sequel, we will analyse the progress of the world record of men, and the world record of women is left for your own analysis.
A simple prediction method is to draw a straight line through the observed data points and to extrapolate this line into future years. This method is known as regression. The general formula for a straight line is y = a + b×, where in our case y = TIME and × = YEAR. The general formula for the regression line y = a + b× for a set of n observed data points (Xk, Yk) is
and a = Y - BX
Here and denote the sample mean of the n observations of respectively Xk and Yk. For the world records of men, we have n = 12 observations, and from the table with world record data we get the values = 1995.167 and = 9.814. Use these values to compute the 12 values of and of (Xk - ×) , and of (Yk - Y) obtain b = -0.008146 and a = 26.067. So, the regression method gives the following equation for the straight line:
TIME = 26.067 – 0.008146 YEAR
The interpretation of this line is as follows. During the period 1968-2010, the world record decreased on average by about 0.008 second per year, that is, by somewhat less than one hundredth of a second. The next diagram shows this line, together with the data, for men on the left and for women (with their own linear regression line) on the right. The equations in the diagrams are not expressed in calendar time, but in a scaled time defined as (YEAR – 2000)/1000.
It is left as an exercise to show that the above equation for men, in terms of YEAR, translates into the equation shown in the diagram for this scaled time.
2. Check and use the model
The diagram with the data and regression line for men show that the data do not fit perfectly to the line. This is not surprising, as the world records do not come in a perfectly predictable sequence. The diagram shows also that the most recent record, of 9.58 by Usain Bolt, is considerably faster than what would be predicted on the basis of previous records. In this race, Bolt improved the world record by 0.11 seconds, whereas previous record improvements ranged between 0.01 and 0.05 seconds. So, this improvement by more than one tenth of a second was really sensational.
The model can be used to predict the next world record, as follows. The current record is 9.58, and the model predicts that this will be broken when the regression line becomes below the level 9.58. This means that the condition for a new record is that
TIME = 26.067 – 0.008146 YEAR < 9.58.
Solving this inequality for the calendar year, we get YEAR > 2023.9. So, the regression line predicts that the current record could hold until 2024. Although it may seem improbable that a record lasts for 15 years, this is not an exception, as can be seen from the world record for women that has remained the same since 1988.
The above prediction is based on a very simple model. We did not take into account other factors that could affect the world record, such as technological developments (training methods, diets, racing shoes, and so on) and human factors. It may be that Bolt (or someone else) is a very exceptional athlete who can establish a new world record before 2024.
Another shortcoming of the model is that it can not be used for predictions in the very long run. If the regression line is extrapolated very far in the future, the running time would even become negative (after the year 3200 …). The regression line is only an approximation and the improvements of the record will become smaller and smaller. This can be modelled, for example, by considering the percentage time reduction (instead of the time reduction in seconds) that is gained in each record.
A final question is whether the amount of data suffices to be confident about the model. The employed data set is restricted to world records that are measured with electronic timing. The records before 1968 could be added to obtain more data, but it is questionable if these data are useful for predicting the future. Today, top runners are professionals who devote their life to their sport, whereas in old times running was no more than a serious hobby. For example, at the 1948 Olympic Games in London, Fanny Blankers-Koen from the Netherlands won four gold medals, among others on the 100 meters. She was called ‘The Flying Housewife’, as it was her job to run the household of her family.