For this research study, we are going to use quantitative research methodology.The quantitative approach entails a research design whose methods put emphasis on objective measurements and analysis of data collected through polls, questionnaires and surveys (Daintith, 2004). Through quantitative research, the relationship between one thing (the independent variable) and another (the dependent or outcome variable) can be established. In this study, the dependent variable is the academic grades while the independent variable is the number of hours of exercise. This is because we expect to establish how varying the number of hours of exercise will ultimately affect the grades of the student.

The logic behind this research design was convergent reasoning rather than divergent reasoning. Convergent reasoning refers to a process of starting from general data and slowly narrowing down to a specific conclusion from the data collected and analyzed. It also entails the processes of spontaneously generating ideas related to the research problem; this may be described as a free-flowing manner of idea generation

The process of conducting the research was based on the basic characteristics of quantitative research. First of all, structured instruments needed to be developed. In this step of the research, structured questionnaires and interview questions were developed. The questions in the questionnaires and interview template were based on the objectives of the research study. Most of the questions had a set of control which was in the multiple choices provided for each of the questionnaires. By using these structured instruments, we could find out the number of hours that students engaged in physical exercise. Secondly, we needed to use the secondary data source to collect data regarding the student who was part of the study.

Data collection, analysis, and representation

Quantitative research requires the collection of statistical data which can be summarized and analyzed by statistical methods. For this research, we need to collect two data sets from a sample population. Our sample population was a class of 25 students; a number which was sufficient to give us reliable results with regard to our study.

The first data set that was collected was the number of hours that each of the students spent exercising during the school day and when not in school. This was through the questionnaires earlier developed. After collecting the first data set, data on the academic performance of the students needed to be gathered; this data formed the second data set for the investigations. This was accomplished by accessing the school databases containing the performance records of each of the students; a secondary source of data.

Once the information was collected, it was tabulated to represent the average number of hours spent exercising and the corresponding average academic grades of the students. For analysis, the mean, median and mode of the data were established. There was also need to establish the least square of regression for the data and correlation coefficient for the given data so as to establish the relationship between the variable and determine the strength of the relationship. Statistical graphs bar and linear line graphs were used for data representation.

Mathematical Investigation

The sample data collected was tabulated as follows;

Table 1: Data collected on number of hours of exercise and student G.P.A

Number of hours of exercise Number of students (data frequency) Average academic performance (G.P.A scores)

0 3 2.3

1 10 3.0

2 7 3.3

3 3 3.7

4 2 4.0

This data was then represented in a graphical manner using a line graph. Plotting the line graph enabled the establishment of the relationship between these two variables of this study. This is because these graphs give a visual representation of the proportionality between data sets.

Plotting the values from the above table we get a straight line graph. From the graph above we see that the number students G.P.A increases with the increases as the number of hours of exercise. This displays direct proportionality between the two variables. The academic performance of a student (represented by the G.P.A) is directly proportional to the number of hours of exercise. This can be interpreted as follows; the more a student involves himself in physical exercise the more likely that he is going to perform better in class. On the other hand, a student that takes little priority in physical activity is likely to exhibit poor or less than average academic performance.This can is also hinted by various aspects of the collected data such as the mode and median. The median of the data being one hour of exercise, it is evident that students that have at least an hour of exercise have a G.P.A of above average (3.0). The mode of the data is two hours; students in this category have even a higher G.P.A of 3.3

To further show the relationship between the two data sets; we are going to look at the least squares of regression for this data collected.Least square regression is essential in identifying the relationship between independent and dependent variables (x and y respectively). In this study, as earlier stated, the G.P.A is the dependent variable while the number of hours of exercise is the independent variable. Least squares of regression is given by the formula y-y=SxySx2(x-x) where Sxy=xyn-xy and Sx2=x2n-x2 Table2. Least squares of regression table:

x y xy x2

0 2.3 0 0

1 3.0 6.0 4

2 3.3 9.9 9

3 3.7 14.8 16

4 4.0 20.0 25

= 10 = 16.3 = 50.7 = 54

x = 2.0 y = 3.26 xy = 6.52 x2 = 10.8

Sxy=xyn-xySxy = (50.7 5) - 6.52

= 10.15 6.52

= 3.63

Sx2=x2n-x2= 54/5 - 10.8

= 3.28 10.8

= -7.52

y-y=SxySx2(x-x)y - 3.26 = 3.63/-7.52 (x - 2.0)

y - 3.26 = -0.50 (x -2.0)

y - 3.26 = -0.50x -1.0

y = - 0.5x -1.0 + 3.26

y = - 0.5x +2.26

Having established the nature of relationship between the dependent and independent variable using the least square regression it is essential to establish the strength of this relationship. To achieve this there is need to calculate the Pearsons Correlation Coefficient; a great indicator of strength of variables relationship. The Pearsons Correlation Coefficient is given by the formula:

r=SxySxSy Where Sx=x--x2nSy=y--y2/n and Sxyis the covariance xyn-xyTable 3: Values of Correlation Coefficient

x y (x-x)2 (y-y)2

0 2.3 4 0.9216

1 3.0 1 0.0676

2 3.3 0 0.0016

3 3.7 1 0.1936

4 4.0 4 0.5476

= 10 = 16.3 = 10 = 1.732

x = 2.0 y = 3.26 Sx=x--x2n= 10/5

= 2

= 1.3142

Sy=y--y2n= 1.732/5

= 0.3464

= 0.4886

r=SxySxSy= 3.63 / 1.3142 * 0.4886

= 5.6531

r2 = 0.31708

The values of correlation coefficient usually range from -1 to +1. As the value of correlation coefficient approaches zero, the more its data points vary from the line of best fit. Moreover, different values of correlation coefficient reflect the strength of the relationship between variables. For instance, a high correlation is shown by values between 5 and 1.0, medium correlation is indicated by values between 3 and 4.9, and low correlation is shown by a value ranging from -1.0 to 2.9. For this data set, the correlation coefficient is 5.6531. This shows that the relationship between the G.P.A of a student (representing the students performance) and the number of hours of exercise is strong.

Discussion

From the results of the mathematical investigation of the study, it is evident that a students academic performance is significantly influenced by the number of hours they engage themselves in physical exercise. The results show an increase in the G.P.A of students as the duration they spend exercising is increased. This can be attributed to the fact that daily physical exercise has a direct impact on the brain development and behavior of students. Physical activity has a significant effect on the cognition of highly plastic developing brains. So how does physical activity enhance brain development and cognition process? First of all, there is increased the flow of oxygen to the brain during exercise; a factor that enhances cognition process. Secondly, due to exercise, the brain of an individual develops more neurotransmitter; another step toward enhanced brain development and cognition. Third, physical exercise increases neurotrophins that are derived from the brain. Neurotrophins are essential in neuronal differentiation support and ensuring neuronal survival during brain differentiation. They are also responsible for ensuring that neurons responsible for higher thinking, memory, and learning survive during brain development(Sparkpe.org, 2016). It is through these three activities that the brain improves brain activities amongst young individuals. Taking into consideration that brain development and cognition processes are essential in the learning process, physical activity, therefore, enhance...

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