1. What is the relationship between the weight and the mile run time?<br />IB Math Studies Internal Assesment<br />International School Bangkok<br />Teacher: Mr. DeMille<br />What is the relationship between the weight and the mile run time?<br />Introduction<br />Although there are diverse physical conditions among students in the school, there are always students who run long distance in short time. No matter if they are fit or athletic, there are always fast in mile run. For example, one boy in my P.E. class is the slowest runner in the 100meter sprint run. He is neither tall nor athletic, but always gets the first place in the mile run. However, he is very thin compare to other boys. He weighs 54 kilograms. Usually in the mile run, students who get short times are quite thin. Also, marathon runners in Olympics are mostly thin compare to sprinters. There are no fat people who run long distance in short time in either school or Olympics. Therefore, the question remains; does the weight of a human body affect the long distance running?<br />Statement of Task<br />The main purpose of this investigation is to determine whether there is a relationship between the weight and the mile run time. Weight is the force with which a body is attracted to Earth or another celestial body, equal to the product of the object's mass and the acceleration of gravity. Mile run is a physical performance test done in every international school in order to measure how fast the runner can run a mile. In order to investigate this question, data was collected in from 30 male high school students from three international schools in Bangkok including ISB, Ekkamai International School, and American School of Bangkok.<br />Plan of Investigation<br />Mathematical processes including; standard deviation, least square regressions, correlation coefficiency, and the Chi-square test were used to investigate the data.<br />Data<br />Table 1: weight and mile run time for 30 international high school students<br />(ISB) No.WeightTime / Seconds(EIS) No.WeightTime/ Seconds(ASB) No.WeightTime/ Seconds1746:50/41011526:40/40021627:10/4302697:20/44012666:30/39022676:30/3903596:10/37013828:10/49023756:50/4104849:00/54014799:10/55024707:00/4205637:00/42015606:30/39025929:40/5806676:20/38016596:10/3702610210:40/6407606:50/41017625:50/35027585:50/3508857:00/42018687:40/46028636:00/3609555:40/34019717:30/45029686:50/41010605:50/35020686:20/38030707:10/430Table 1: Table 1 shows the data of weight and mile run time collected from 10 high school students from ISB, 10 from EIS, and 10 from ASB. The mile run time is rounded up to 10s of seconds.GraphExcel generated scatter plot of the collected data<br />Graph 1: Graph 1 slightly shows that the heavier the student is the larger the number of the time.<br />Standard Deviation Calculations<br />Standard deviation shows how much variation there is from the average of the particular variables (in this case, of weight and mile run time).<br />The formula is given:<br /> Sx=x2n-x2 Sy=y2n-y2<br />Sx=11.2<br />11.2 is the standard deviation of X, which is weight.<br />Sy=71.7<br />71.7 is the standard deviation of Y, which is mile run time.<br />Least Squares Regression<br />Least squares regression shows the relationship between independent variable X, and the dependent variable Y.<br />The formula is given:<br />y-y=SxySx2x-x where Sxy=xyn-xy<br />Sxy=89852030-29276.7<br />Sxy=674<br />Therefore:<br />y-424.3=674125x-69<br />y-424.3=5.4x-69<br />y=5.4x+51.7<br />Pearson’s Covariance Coefficiency<br />The equation for the correlation coefficient is r=SxySxSy Where Sx=x-x2n , Sy=y-y2n and<br /> Sxy is the covariance xyn-xy.<br />r=674803<br />r=0.84<br />r2=0.7<br />Graph2<br />Excel generated scatter plot of the collected data with calculation program of the least squares regression line.<br />Graph 2: Graph 2 shows that there are correlation between two variables by telling that the correlation is 0.8608 which is close to 1.<br />Discussion<br />Data Interpretation<br />Firstly, the graph of this data clearly shows that it is a positive linear correlation. However, it appears to be slightly weak. Pearson’s correlation coefficiency of this data was calculated r2as 0.7, which is not far from 1.0. This indicates that the valuables actually are showing correlation. <br /> Secondly, the chi-square test shows that….<br /> Although weights and mile run times are showing correlation, it must be known that it doesn’t always correlates. For example, if the person is very thin, but is heavy because of his height, he might still be able to run in short time.<br />Limitations<br />Since each of the students had different conditions when they ran, it is hard to rely on this data. For example, even if the student can run fast, if he was on bad condition, it would affect the result visibly.<br />Secondly, only concentrating on weight is not as good way of investigating against the mile run. Since each student has different heights, weight might not be affecting the performance. For example, if the student who weighs 70kg but 180cm, his physical appearance must be skinny. Thus, only concentrating on the weights is the limitation of this investigation.<br />Lastly, the mathematical mistakes in this investigation are the limitation. Since this investigation is formed in step-by-step, one mathematical mistake could affect the whole results of the data during the calculations.<br />Conclusion<br />Despite the limitations mentioned previously, the results of the investigation still shows that there is a relationship between weight and mile run time of the students. It is a common sense that obese people take longer to run a mile than other people. However, this investigation showed that not only the obese people but also heavy people take longer on mile run. It is expected that consume of energy affected by gravity is answer of this investigation.<br />