Use Mathematics in A Range Of Contexts


 



 


 


 


 


 


 


 


 



 

Introduction


 


            Nowadays, swimming as sport is known to everyone. Regardless of gender you have, swimming abilities of men and women are seem to be equal. Sexual differences or gender variations is not a matter of pre-existing categories with set contents, but is an interval or gap — a radical difference — between the sexes’ experiences and knowledge’s. It does not fit clearly into the dualism of nature and culture (Archer, S. 1989). Some may argue that gender have always been at the centre of various global discussions. The distinction between women and men is of great interest to researchers. Biologically speaking, women and men differ from each other. Each group has their own physical make up that is distinguishable from one another. Men are basically perceived as physically stronger and robust while women are more gentle and slender. The physical strength brought about by the anatomical characteristics of men typically becomes the basis for gender inequality. Since men are more muscular and stronger, women are put into a subordinate status. However, the essence of gender extends beyond biological evidence. Gender refers to the meanings that are attached to those differences within a culture. Sex is male and female; gender is masculinity and femininity or what it means to be a man or a woman.  But in swimming, are they different? Who excel the most? Basically, this paper will be discussing the previous swimming records of both men and women from different years and location of competition.  Comparison and prediction of future swimming records will be also shown in this paper. 


Swimming Records of Women


 


            In previous years, women are joining in different swimming events.  These events are usually joined by different participants around the globe. As seen in figure 1, most of women participants in swimming event and considered in this paper are from DDR or the known German Democratic Republic (Deutsche Demokratische Republik). Basically from year 1966 up to year 2008, the overall women participants in swimming from DDR are 52.38%. This was also followed by participants from USA which is about 28.57%.  As shown in the figure, women participants from China and Australia had only 9.52% of representatives for each.


            In figure 2, the frequencies of participation of women with respect to year’s range are shown.  As indicated in this figure, most women or 67% of the overall women participated in swimming competition occurred in year 1973 to 1981.  There are also expressive numbers of women participants in years 1964-1972 i.e. 19%. Followed by 9% in year 1991-1999 and 5% in year 2000 up to 2008. However, in 1982 to 1990 the record shows that no women joined in any swimming event from the selected swimming records.


            With regards to the swimming records of women, Figure 3 illustrates the average swimming speed of women.  As the records shows, women shows expressive performance in year 1991-1999 with an average swimming speed of 120.135 seconds in 200 metres Individual Medley.  As illustrated, the average swimming records of women in year 1964-1972 in 200 metres Individual Medley is 146.21, 137.028 seconds in year 1973-1981 and 128.92 for years 2000-2008.  Since the selected records used in this paper indicates that no women joined in 1982-1990 swimming events, then their average swimming records for the said years is not recorded.          


 


Men Swimming Records



            Contrary to the swimming records of women, men participants in global swimming competition are usually from USA.  As shown in figure, majority or 61% of the total men swimming participants are from USA. Followed by 16% from Canada.  On the other hand, there are also quite good numbers of participants joined in global swimming competition from Hungary which was comprises of 10% of overall representatives around the globe.  The record also shows that 7% of it are from Sweden, 3% from Finland, and another 3% from Great Britain.


            Men participation in swimming competition throughout the years are expressive.  As indicated from the records, 26% of overall men swimming representatives joined the competitions from years 1964-1972.  Moreover, another 26% of them joined the competition in years 1973-1981.  There were also 23% of men participants joined in 2000-2008 200 metres Individual Medley. 19% from 1982-1990 and 6 from 1991-1999.


            As shown in figure 6, the average swimming speed of men throughout the years shows some consistency.  As the year pass by, the average swimming speed of men increases.   For the record, men average swimming speed for 1964-1972 was only 129.9 and now for 2000 to 2008, men speed increases to an average of 116.15 seconds in 200 metres Individual Medley.


Comparison and Prediction of Swimming Records


 


T-test


            To determine if there are significant differences between the swimming performance of men and women throughout the years, the use of independent sample t-test was used.


Table 1. Descriptive



 


This table displays the number of cases, mean value, standard deviation, and standard error for the test variable(s) within categories defined by the grouping variable. Since the Independent Samples T Test procedure compares the two group means, it is useful to know what the mean values are.  Basically, the mean and standard deviation results will be used in coefficient of variation analysis.


Table 2. Independent Sample T-test



 


Actually, the Independent-Samples T-Test procedure compares means for two groups of cases (Walpole,R.E., Myers,R.H.,  Myers,S.L.,& Ye. K. 2002). The mean values for the two groups are displayed in Table 1. Since the significance value for the Levene test is high (i.e. greater than 0.05), then we have to use the results that do not assume equal variances for both groups.


Basically, a low significance value for the t-test (typically less than 0.05) indicates that there is a significant difference between men and women means. Since the confidence interval for the mean difference does not contain zero, this also indicates that the difference is significant (Walpole, et al 2002). Since the computed sig. vale for gender variable with respect to their swimming speed in 200 metres Individual Medley is 0.000, then we may say that the swimming speed in 200 metres Individual Medley of men through the years is different to the speed of women.


            From the series of statistical tests and analysis, it is found out that there is an existing gender-based difference in terms of swimming speed in 200 metres Individual Medley.  To identify which gender dominates in terms of swimming speed, the use of coefficient of variation analysis will be considered.


 


Coefficient of Variation Analysis


To find if there is significant evidence that the swimming speed of men is lower than in the women, we have to compute the Coefficient of Variation of the two groups of data. Thus, , where s= sample standard deviation and  is the sample mean.  The details in table 1 such as mean and standard deviation was used (Walpole, et al 2002).


 


 


 


 


Computation:


Male:                                                               Female:


               


      


                   


 


            As shown in the computation, the male group is more variable in comparison to the female group, thus we may say that there is significant evidence that the swimming speed of men from previous years of performance is better than women.  The results also show consistency which is an indication of possibilities of easy detection of the average speed of a swimmer if the competition year is given.


 


Linear Regression                    


Suppose we are trying to predict some continuous Y variable (competition years) from X (swimmers speed) and we obtain the scatter plot to the right. We wish to construct the “best fitting” prediction line which captures the linear relationship between X and Y and allows us to predict Y from a given X score (Walpole, et al 2002). We wish to form this prediction line such that we make as few prediction errors as possible. An error in prediction is defined as the difference between each subject’s actual score and the predicted Y score obtained from X via the prediction line (Walpole, et al 2002).



            The regression line which minimizes errors of prediction for the whole sample will have the following formula, including a breakdown for the slope (b) and the intercept (a) (Walpole, et al 2002):



            Note that the slope of the regression line is highly related to the correlation coefficient. Thus, we can obtain the prediction line from each X value via:



            Furthermore, if X and Y are expressed as standard scores (i.e., we have converted raw scores for X and Y to z scores), then using a little algebraic massaging:



 


            With this information and in order to compute the least regression line we have to use the following table and formula:


y


Years


y


Time in Seconds


xy


x2


y2


 


 


1997


129.72


259050.84


3988009


16827.2784


 


1992


131.35


261649.2


3968064


17252.8225


 


1981


131.73


260957.13


3924361


17352.7929


 


1980


133


263340


3920400


17689


 


1980


133.69


264706.2


3920400


17873.0161


 


1978


134.07


265190.46


3912484


17974.7649


 


1978


135.09


267208.02


3912484


18249.3081


 


1977


135.85


268575.45


3908529


18455.2225


 


1977


135.95


268773.15


3908529


18482.4025


 


1977


136.96


270769.92


3908529


18758.0416


 


1976


137.14


270988.64


3904576


18807.3796


 


1976


138.3


273280.8


3904576


19126.89


 


1975


138.83


274189.25


3900625


19273.7689


 


1974


138.97


274326.78


3896676


19312.6609


 


1973


140.51


277226.23


3892729


19743.0601


 


1973


143.01


282158.73


3892729


20451.8601


 


1972


143.07


282134.04


3888784


20469.0249


 


1967


146.1


287378.7


3869089


21345.21


 


1967


147.5


290132.5


3869089


21756.25


 


1966


147.8


290574.8


3865156


21844.84


 


2008


114.8


230518.4


4032064


13179.04


 


2007


114.98


230764.86


4028049


13220.4004


 


2006


115.84


232375.04


4024036


13418.9056


 


2003


115.94


232227.82


4012009


13442.0836


 


2003


116.04


232428.12


4012009


13465.2816


 


2003


117.52


235392.56


4012009


13810.9504


 


2003


117.94


236233.82


4012009


13909.8436


 


1994


118.16


235611.04


3976036


13961.7856


 


1991


119.36


237645.76


3964081


14246.8096


 


1989


120.11


238898.79


3956121


14426.4121


 


1988


120.17


238897.96


3952144


14440.8289


 


1987


120.56


239552.72


3948169


14534.7136


 


1984


121.42


240897.28


3936256


14742.8164


 


1986


121.42


241140.12


3944196


14742.8164


 


1982


122.25


242299.5


3928324


14945.0625


 


1981


122.78


243227.18


3924361


15074.9284


 


1980


123.24


244015.2


3920400


15188.0976


 


1979


123.29


243990.91


3916441


15200.4241


 


1978


125.24


247724.72


3912484


15685.0576


 


1977


125.31


247737.87


3908529


15702.5961


 


1975


126.32


249482


3900625


15956.7424


 


1974


126.32


249355.68


3896676


15956.7424


 


1974


126.8


250303.2


3896676


16078.24


 


1972


127.17


250779.24


3888784


16172.2089


 


1972


129.3


254979.6


3888784


16718.49


 


1970


129.3


254721


3880900


16718.49


 


1970


129.5


255115


3880900


16770.25


 


1969


129.6


255182.4


3876961


16796.16


 


1968


130.6


257020.8


3873024


17056.36


 


1967


131.3


258267.1


3869089


17239.69


 


1966


132.4


260298.4


3865156


17529.76


 


Σx=101042


Σy=6573.62


Σxy=13019694.93


Σx2=200193120


Σy2=851377.5818


 


 


The regression line formula: y= a+bx,  and


Using the table and formula we have:


 





 





 


From the computed a and b, the regression line of the given data becomes:  or simply


Let’s say, the predicted swimming speed of a swimmer in 2009 would be  = 131.39 seconds in 200 metres Individual Medley.


 


 


Conclusion


 


                 From the results of analysis, we can conclude that men dominates women in 200 metres Individual Medley in terms of swimming speed throughout the years.


 


References:


Archer, S. (1989). Gender differences in identity development: Issues of process, domain, and timing. Journal of Adolescence, 12, 117-138.


 


Walpole,R.E., Myers,R.H.,  Myers,S.L.,and Ye. K. (2002) Probability and Statistics for Engineers and Scientists (Seventh   edition), Prentice-Hall.


 


 



Credit:ivythesis.typepad.com


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