Effect of Short Range Correlation

The effect of short range correlation on the nuclear charge density distribution, elastic and inelastic electron scattering coulomb form factors of 16O nucleus
Abdullah S. Mdekil
Abstract
The effect of the short range correlation on the charge density disribution, elastic electron scattering form factors and inelastic Coulomb form factors is studied for the two excited states (6.92 and 11.52 MeV) in is analyzed. This effect (which depends on the correlation parameter) is inserted into the ground state charge density distribution through the Jastrow type correlation function. The single particle harmonic oscillator wave function is used with an oscillator size parameter The parameters and are considered as free parameters, adjusted for each excited state separately so as to reproduce the experimental root mean square charge radius of In inelastic coulomb (longitidinal) form factors of 16O, two different models are employed for . In the first model (model A), is considered as a closed shell nucleus. Here, the model space in does not contribute to the transition charge density, because there are no protons outside the closed shell nucleus . In the second model (model B), the nucleus of is assumed as a core of with 2 protons and 2 neutrons move in and model space. It is found that the introduction of the effect of short range correlations is necessary for obtaining a remarkable modification in the calculated inelastic Coulomb form factors and considered as an essential for explanation the data amazingly throughout the whole range of considered momentum transfer.
Keywords: charge density distribution, elastic charge form factors, inelastic longitudinal form factors, short range correlation.
1-Introduction
Electron scattering provides more accurate information about the nuclear structure for example size and charge distribution. It provides important knowledge about the electromagnetic currents inside the nuclei. Electron scattering have been provided a good test for such evaluation since it is sensitive to the spatial dependence on the charge and current densities [1, 2, 3].
Depending on the electron scattering, one can distinguish two types of scattering: in the first type, the nucleus is left in its ground state, that is called “elastic electron scattering” while in the second type, the nucleus is left on its different excited states, this is called “inelastic electron scattering” [4, 5].
In the studies of Massen et al. [6-8], the factor cluster expansion of Clark and co-workers [9-11] was utilized to derive an explicit form of the elastic charge form factor, truncated at the two-body term. This form, which is a sum of one- and two-body terms, depends on the harmonic oscillator parameter and the correlation parameter through a Jastrow-type correlation function [12]. This form is employed for the evaluation of the elastic charge form factors of closed shell nuclei and in an approximate technique (that is, for the expansion of the two-body terms in powers of the correlation parameter, only the leading terms had been kept) for the open and shell nuclei. Subsequently, Massen and Moustakidis [13] performed a systematic study of the effect of the SRC on and shell nuclei with entirely avoiding the approximation made in their earlier works outlined in [6-8] for the open shell nuclei. Explicit forms of elastic charge form factors and densities were found utilizing the factor cluster expansion of Clark and co-workers and Jastrow correlation functions which introduce the SRC. These forms depends on the single particle wave functions and not on the wave functions of the relative motion of two nucleons as was the case of our previous works [14-20] and other works [6,21,22].
It is important to point out that all the above studies were concerned with the analysis of the effect of the SRC on the elastic electron scattering charge form factors of nuclei.
There has been no detailed investigation for the effect of the SRC on the inelastic electron scattering form factors of nuclei. We thus, in the present work, perform calculations with inclusion this effect on the inelastic Coulomb form factors for closed shell nucleus. As a test case, the is considered in this study. To study the effect of SRC (which depends on the correlation parameter on the inelastic electron scattering charge form factors of considered nucleus, we insert this effect on the ground state charge density distribution through the Jastrow type correlation function [12]. The single particle harmonic oscillator wave function is used in the present calculations with an oscillator size parameter The effect of SRC on the inelastic Coloumb form factors for the two excited states (6.92 and 11.52 MeV) in is analyzed.
2. Theory
Inelastic electron scattering longitudinal (Coulomb) form factor involves angular momentum and momentum transfer and is given by [23]
(1)
where and are the initial and final nuclear states (described by the shell model states of spin and isospin ), is the longitudinal electron scattering operator, is the center of mass correction (which removes the spurious states arising from the motion of the center of mass when shell model wave function is used), is the nucleon finite size correction and assumed to be the same for protons and neutrons, A is the nuclear mass number, is the atomic number and is the harmonic oscillator size parameter.
The form factor of eq.(1) is expressed via the matrix elements reduced in both angular momentum and isospin [24]
(2)
where in eq. (2), the bracket ( ) is the three- symbol, where and are restricted by the following selection rule:

(3)
and is given by
The reduced matrix elements in spin and isospin space of the longitudinal operator between the final and initial many particles states of the system including configuration mixing are given in terms of the one-body density matrix (OBDM) elements times the single particle matrix elements of the longitudinal operator [25]
(4)
where and label single particle states (isospin included) for the shell model space. The in eq. (4) is calculated in terms of the isospin-reduced matrix elements as [26]
(5)
where is the isospin operator of the single particle.
(6)
The model space matrix element, in eq. (6), is given by
(7)
where is the spherical Bessel function and is the model space transition charge density, expressed as the sum of the product of the times the single particle matrix elements, given by [26].
(8)
Here, is the radial part of the harmonic oscillator wave function and is the spherical harmonic wave function.
The core-polarization matrix element, in eq. (6), is given by
(9)
where is the core-polarization transition charge density which depends on the model used for core polarization. To take the core-polarization effects into consideration, the model space transition charge density is added to the core-polarization transition charge density that describes the collective modes of nuclei. The total transition charge density becomes
(10)
According to the collective modes of nuclei, the core polarization transition charge density is assumed to have the form of Tassie shape [27]
(11)
where is the proportionality constant given by [14]
(12)
which can be determind by adusting the reduced transition probability to the experimental value, and is the ground state charge density distribution of considered nuclei.
For the ground state charge densities of closed shell nuclei may be related to the ground state point nucleon densities by [28, 29]
(13)
in unit of electronic charge per unit volume (e.fm-3).
An expression of the correlated density (where the effect of the SRC’s is included), consists of one- and two-body terms, is given by [13]
(14)
where is the normalization factor and is the one body density operator given by
(15)
The correlated density of eq. (14), which is truncated at the two-body term and originated by the factor cluster expansion of Clark and co-workers [10-12], depends on the correlation parameter through the Jastrow-type correlation
(16)
where is a state-independent correlation function, which has the following properties: for large values of and for It is so clear that the effect of SRC’s, inserted by the function becomes large for small values of SRC parameter and vice versa.
The one-body term, in eq. (14), is well known and given by
(17)
where is the occupation probability of the state and is the radial part of the single particle harmonic oscillator wave function.
The two-body term, in eq. (14), is given by [13]
(18)
where
(19)
The form of the two-body term is then originated by expanding the factor in the spherical harmonics and expressed as [13]
(20)
where
(21)
and is the Clebsch-Gordan coefficients.
It is important to point out that the expressions of eqs. (17) And (20) are originated for closed shell nuclei with where the occupation probability is 0 or 1. To extend the calculations for isotopes of closed shell nuclei, the correlated charge densities of these isotopes are characterized by the same expressions of eqs. (17) and (20) (this is because all isotopic chain nuclei have the same atomic number but this time different values for the parameters and are utilized.
The mean square charge radii of nuclei are defined by
(22)
where the normalzation of the charge density distribution is given by
(23)
3-Results and discussion
The ground state CDD is calculated by eq.(13) together with eqs. (14), (17) and (20). The calculated CDD without (with) the effect of the SRC [i.e., when the correlation parameter is obtained by adjusting only the parameter (the two parameters and ) so as to reproduce the experimental root mean square (rms) charge radii of nuclei under study. The elastic electron scattering charge form factors which is simply the Fourier transform of the ground state CDD.
In Fig. 1, we compare the calculated CDD [Fig. 1(a)] and elastic charge form factors [Fig. 1(b)] of with those of experimental data (the open circles). In Fig. 1, we compare the calculated CDD [Fig. 1 (a)] and elastic charge form factors [Fig. 1 (b)] of with those of experimental data (the open circles). The dashed curves are the calculated results without the inclusion of the effect of the SRC obtained with and fm. The solid curves are the calculated results with including the effect of the SRC obtained with fm-2 and fm. It is important to point out that the parameters and employed in the calculations of the dashed and solid curves are chosen so as to reproduce the experimental rms charge radius of Fig. 1 (a) illustrates that the calculated CDD of the dashed curve (without the effect of the SRC) is in such a good agreement with that of the experimental data, and the solid curve (with the effect of the SRC) is not in such a good agreement with that of the experimental data, especially in the central region ( fm) of the distributions. The inclusion of SRC has the feature of reducing the central region of the distribution as seen in the solid curve of this figure. Inspection to the Fig. 1 (b) gives an indication that the solid curve is better describing the experimental data than that of the dashed curve, particularly in the region of momentum transfer fm-1. The rms charge radius calculated with the above values of and is 2.621 fm, which is less than the experimental value by 0.097fm, which corresponds to a decrease of nearly 3.6 % of the experimental value.

Fig. 1. The calculated CDD and elastic charge form factors are compared with those of experimental data. The dashed curve corresponds to the values for the parameters and fm, the solid curve corresponds to the values for the parameters fm-2 and fm while the open circles and the triangles in Figs. 1 (a) and 1 (b) are the experimental data taken from [30] and [31], respectively.
The effect of the SRC on the inelastic Coulomb form factors is studied for the two excited states (6.92 and 11.52 MeV) in. Core polarization effects are taken into consideration by means of the Tassie model [eq. (11)], where this model depends on the ground state charge density distribution. The proportionality constant [eq. (12)] is estimated by adjusting the reduced transition probability to the experimental value. The effect of the SRC is incorporated into the ground state charge density distribution through the Jastrow type correlation function [12]. The single particle harmonic oscillator wave function is employed with an oscillator size parameter
The charge density distribution calculated without the effect of the SRC depends only on one free parameter (namely the parameter), where is chosen in such away so as to reproduce the experimental rms charge radii of considered nuclei. The charge density distribution calculated with the effect of the SRC depends on two free parameters (namely the harmonic oscillator size parameter and the correlation parameter), where these parameters are adjusted for each excited state separately so as to reproduce the experimental rms charge radii of considered nuclei.
Two different models are employed for. In the first model (model A), is considered as a closed shell nucleus. In this model, the proton occupation probabilities in are assumed to be and Here, the model space in does not contribute to the transition charge density [i.e. ], because there are no protons outside the closed shell nucleus . Accordingly, the Coloumb form factors of come entirely from the core polarization transition charge density. In the second model (model B), the nucleus of is assumed as a core of with 2 protons and 2 neutrons move in and model space. In this model, the proton occupation probabilities in are assumed to be and Here, the total transition charge density [eq. (10)] comes from both the model space and core polarization transition charge densities. The OBDM elements of are generated, via the shell model code OXBASH [32], using the REWIL [33] as a realistic effective interaction in the isospin formalism for 4 particles move in the and model space with a core.
In Table 1, the experimental excitation energies (MeV), experimental reduced transition probabilities (fm) and the chosen values for the parameters and for each excited state (used in the calculations of model A and B) in and are displayed. The root mean square (rms) charge radius calculated in both models with the effect of SRC is also displayed in this table and compared with that of experimental result. It is evident from this table that the values of the parameter employed for calculations with the effect of SRC are smaller than that of without SRC ( fm) . This is attributed to the fact that the introduction of SRC leads to enlarge the relative distance of the nucleons (i.e., the size of the nucleus) whereas the parameter (which is proportional to the radius of the nucleus) should become smaller so as to reproduce the experimental rms charge radius of the considered nuclei.
Inelastic Coloumb form factors for different transitions in are displayed in Figs. 1 and 2. The calculated inelastic form factors obtained with model A are shown in the upper panel [Figs. 1(a)-2(a)] of the above figures whereas those obtained with model B are shown in the lower panel [Figs. 1(b)- 2(b)] of the above figures. It is obvious that all transitions considered in, presented in the above figures, are of an isoscalar character. Besides, the parity of them does not change. Here, the calculated inelastic form factors are plotted versus the momentum transfer and compared with those of experimental data. The dashed and solid curves are the calculated inelastic Coloumb form factors without and with the inclusion of the effect of the SRC, respectively. The open symbols are those of experimental data taken from [34, 35].
Table1. The experimental excitation energies and reduced transition probabilities, the chosen values for and as well as the rms charge radius calculated with the effect of the SRC of 16O.

The Correlation of Sleep Quality, Perceived Stress, and Academic Performance on University Students

The purpose of this study conducted is to observe whether the quality of sleep and stress have any effect on the academic performance of university students.

The 2013 study conducted by Calkins, Hearon, Capozzoli, and Otto’s purpose was to figure out how environmental factors-such as insomnia related distress can predict the outcome of sleep quality. Calkins et al. (2013) investigated the connection between neuroticism on sleep disruption, faulty assumptions about sleep, and anxiety sensitivity. This study used 118 undergraduate students from Boston University through advertisements who were enrolled in a psychology course, and in return were given credit towards there grade for participating (Calkins et al., 2013). Calkins et al. (2013) used professional survey software called SurveyMonkey to gather information pertaining to the four scales to measures; Dysfunctional Beliefs and Attitudes about Sleep Scale (DBAS-16), Anxiety Sensitivity Index (ASI), Neuroticism, Extraversion, Openness to experience Five-Factor Inventory (NEO-FFI), and the Pittsburgh Sleep Quality Sleep Index (PSQI). Calkins et al. (2013) hypothesized that higher AS and neuroticism, and greater faulty opinions about sleep would significantly result in decreased quality of sleep.

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Overall, Calkins et al. (2013) found that women scored significantly higher than men. The results found that there is a significant difference based on ethnicity on DBAS and neuroticism scores, with Latino, African American, and Hispanic participants having relatively lower scores. On the PSQI scale sleep disruption is positively correlated with the ASI score, DBAS, and neuroticism (Calkins et al., 2013). Also, sleep quality and daytime dysfunction subscale were connected with ASI social and mental concerns, DBAS, and neuroticism (Calkins et al., 2013). The sleep latency subscale was only associated with neuroticism and DBAS, while sleep duration and sleep efficiency were found not to be associated with any variable (Calkins et al., 2013). The use of sleeping pills was only associated with DBAS (Calkins et al., 2013).

Calkins et al. (2013) findings significantly put an importance on how neuroticism should not be forgotten to be included when discussing the problems related to sleep dysfunction and also contributes to the development of cognitive symptoms like AS-mental incapacitation and may play a role in sleep-related distress and dysfunction.

The 2010 study conducted by Gilbert and Weaver’s purpose was to determine whether the amount and the quality of sleep in non-depressed university students get is associated with poor academic performance. Gilbert and Weaver (2010) hypothesized that low sleep quality and decreased amount of sleep in non-depressed students would have a poor academic performance than those who are well rested. This study included 468 undergraduate students in an introductory course in psychology who were identified as not having depressive symptoms (Gilbert and Weaver, 2010). This study used three forms of measures including; Demographic Survey (DS), the Goldberg Depression Inventory (GDI), and the Pittsburgh Sleep Quality Index (PSQI) (Gilbert and Weaver, 2010).

Gilbert and Weaver (2010) found there to be a significant negative correlation between Global Sleep Quality (GSQ) and Grade Point Average (GPA) in females; which supports their initial hypothesis that low sleep quality and decreased the amount of sleep in non-depressed students to have poorer academic performance. Students that do not have depression with low quality and reduced amount of sleep had significantly lower GPAs (Gilbert and Weaver, 2010).

Gilbert and Weaver (2010) state how the importance of sleep quality is related to academic success and that the quality of sleep should also be looked at alongside the quantity of sleep since findings point out that college student sleep habits are on the low end. Also, even though students GPA may not be effected through sleep deprivation other social, medical, and cognitive detrimental effects (Gilbert and Weaver, 2010).

The 2012 study conducted by Ahrberg, Dresler, Niedermaier, Steiger, and Genzel’s purpose was to determine the relationship between stress, sleep, and academic performance. Ahrberg et al. (2012) investigated the stress levels and sleep quality in students before, during, and after the exams. This study used 144 out of 632 medical students who completed the online surveys that include a 10-point rating stress scale, and the PSQI. The results show that academic performance is associated with sleep quality and the level of stress before the exam, however not during or after the exam (Ahrberg et al., 2012). PSQI before exam results show that the tension before the exam is still associated significantly with grades (Ahrberg et al., 2012). However, it is seen that students with poor sleep quality do not necessarily get bad grades, but in fact, stress plays a significant role in receiving poor grades as well causing one to have poor sleep quality (Ahrberg et al., 2012). Ahrberg et al. (2012) states that it is a cycle that starts with stress you may be afraid of getting bad grades, which then causes one to feel sleepy due to not getting a proper amount or quality of sleep which then causes a lower academic performance, and comes around in full cycle since low-performance rate cause stress to most students.

It is essential to have a somewhat knowledge of the background of the experiment that is being to conduct, to have some information to support your the overall conclusion along with your results. We will be using the Perceived Stress Scale (PSS) and the Pittsburgh Sleep Quality Index (PSQI) to see whether the undergraduate students overall academic performance was affected by the amount of stress and the sleep quality they had. It is predicted that increased levels of stress and PSQI scores will result in the poor quality of sleep and academic performance of undergraduate students compared to students who have lower stress levels. It is also predicted that lower PSQI scores will result in the student’s academic performance to improve overall. This is anticipated because of the three studies by Calkins et al. (2013), Gilbert and Weaver (2010), and Ahrberg et al. (2012) findings that support the stress level and the quality of sleep is negatively correlated with overall academic performance of undergraduate students.

References

Ahrberg, K., Dresler, M., Niedermaier, S., Steiger, A., & Genzel, L. (2012). The interaction between sleep quality and academic performance. Journal of Psychiatric Research,46(12), 1618–1622. https://doi.org/10.1016/j.jpsychires.2012.09.008

Calkins, A. W., Hearon, B. A., Capozzoli, M. C., Michael, W., Calkins, A. W., Hearon, B. A., … Otto, M. W. (2013). Psychosocial Predictors of Sleep Dysfunction : The Role of Anxiety Sensitivity , Dysfunctional Beliefs , and Neuroticism Psychosocial Predictors of Sleep Dysfunction : The Role of Anxiety Sensitivity , Dysfunctional Beliefs , and Neuroticism, 11(2), 133–143. https://doi.org/10.1080/15402002.2011.643968

Gilbert, S. P., & Weaver, C. C. (2010). Sleep Quality and Academic Performance in University Students : A Wake-Up Call for College Psychologists, 24(4), 295–306. https://doi.org/10.1080/87568225.2010.509245

 

Pearson Correlation Investigate The Relationship Between Variable Physical Education Essay

This chapter discusses the results from data collected and analyzed by researcher. Data collected were analyzed and interpreted using Statistical Package for Social Science (SPSS 17.0) software. This chapter will discuss about subject’s age, VO2max, VO2, HR, and RPE. This will use descriptive statistic to interpret the data. This chapter also will test the hypothesis constructed by researcher by using Pearson Correlation to investigate the relationship exist between those variables.
4.1 HYPOTHYESIS
4.1.1 H0 : There is no significant between VO2max and HR among athlete.
H1 : There is significant between VO2max and HR among athlete.
4.1.2 H0 : There is no correlation between RPE and HR among athlete.
H1 : There is correlation between RPE and HR among athlete.
4.1.3 H0 : There is no correlation between VO2max and RPE.
H1 : There is correlation between VO2max and RPE.
4.2 DESCRIPTIVE STATISTICS
As showed at table 1, number of subjects in this test is 21 and their age are between 20-30 years old with mean 23.24 ± 1.868 years. Subjects VO2max are ranged from 34-69 ml.kg.min with mean 52.33 ± 7.716 ml.kg.min. Heart rate result show ranged from 173-200 bpm with mean 187.67 ± 7.220 bpm. Rating of perceived exertion scale score result show ranged from 3-10 RPE with mean 7.24 ± 2.119 RPE. VO2 result show ranged from 2.512-4.290 l.min with mean 3.435 ± 0.476 l.min. Respiratory exchange ratio result show ranged from 0.96-1.21 with mean 1.09 ± 0.066. For subjects body mass index result show ranged 18-33 with mean 22.90 ± 3.330. Lastly, for sbujects physical activity show a result ranged from 1-4 with mean 3.24 ± 1.091.
Table 1. Subjects Descriptive Statistics
N
Minimum
Maximum
Mean
Std. Deviation
AGE
VO2max
HR
RPE
VO2
RER
BMI
Physical Activity
21
21
21
21
21
21
21
21
20
34
173
3
2.512
.96
18
1
30
69
200
10
4.290
1.21
33
4
23.24
52.33
187.67
7.24
3.435
1.09
22.90
3.24
1.868
7.716
7.220
2.119
0.476
0.066
3.330
1.091
4.3 RELATIONSHIP GRAPH
Figure 1 to 20 below show the relationship between all variables for subjects in Bruce Protocol test.
Figure 1. VO2max and VO2 Figure 2. VO2max and HR
Figure 3. VO2max and RER Figure 4. VO2max and RPE
Figure 5. RPE and HR Figure 6. RPE and VO2
Figure 7. Physical activity and VO2max Figure 8. Physical activity and RER
4.4 CORRELATION
As show at table 2 below, the correlation result for subjects in Bruce Protocol test.
In VO2max, there are no significant correlation between HR with the result of r = .120 and the significant value is p = .605. From the result, VO2max result will have no effect on HR if result score for VO2max are increase or decrease.
For significant correlation VO2max between RPE result, there are no significant with r = .188 and significant value is p = .416. This show, increase in VO2max has no relationship with RPE scale score.
There are significant between VO2max and VO2 with result r = .509 and the significant value is p = .018 which show positive correlation. This show, increase in VO2max might be cause of increase in VO2.
For correlation between VO2max and RER, there are no significant correlation with result r = -.056 and the significant value is p = .810 which show a negative correlation. VO2max has no influence in RER and increase in VO2max has no associated with RER result.
There are significant correlation result between VO2max and physical activity with result r = .459 and significant value is p = .036. The relationship between VO2max and physical activity are positive correlation. Increase in VO2max might be cause of increase in physical activity (hour).
There are no significant relationship between HR and RPE with result r = -.089 and significant value is p = .700. These show a negative correlation between HR and RPE. Increase in HR has no influence in RPE scale score.
In correlation between RPE and VO2, there are no significant correlation result with r = .063 and significant value is p = .785. These show increase in RPE has no relationship with VO2 result.
There are no significant correlation between RER and physical activity result with r = -.256 and significant value is p = .263. These show negative correlation between RER and physical activity. Increase in RER has no relationship with increase in physical activity (hour).
Table 2. Correlation
Correlations
VO2max
VO2
HR
RER
RPE
physical activity
VO2max
Pearson Correlation
1
.509*
.120
-.056
.188
.459*
Sig. (2-tailed)
.018
.605
.810
.416
.036
N
21
21
21
21
21
21
VO2
Pearson Correlation
.509*
1
.332
-.077
.063
.315
Sig. (2-tailed)
.018
.141
.740
.785
.164
N
21
21
21
21
21
21
HR
Pearson Correlation
.120
.332
1
.227
-.089
-.059
Sig. (2-tailed)
.605
.141
.323
.700
.799
N
21
21
21
21
21
21
RER
Pearson Correlation
-.056
-.077
.227
1
.111
-.256
Sig. (2-tailed)
.810
.740
.323
.631
.263
N
21
21
21
21
21
21
RPE
Pearson Correlation
.188
.063
-.089
.111
1
.299
Sig. (2-tailed)
.416
.785
.700
.631
.189
N
21
21
21
21
21
21
physical activity
Pearson Correlation
.459*
.315
-.059
-.256
.299
1
Sig. (2-tailed)
.036
.164
.799
.263
.189
N
21
21
21
21
21
21
*. Correlation is significant at the 0.05 level (2-tailed).
4.5 DISCUSSION AND FINDING
4.5.1 HR and RPE
For relationship between HR and RPE show a significant different result in David B. James, et al studies which showed in test 1 and test 2, and test 2 and test 3. Studies used only 8 male subjects which are runner athlete and average age of 25 ± 6 years old.
However, present finding show that correlation between HR and RPE has no significant result. Study support by Danielle M.L, et al, showed no significant relationship between RPE and HR with result p = >0.05. Present study show most of subject age at 23 years old and this show their maximal heart rate is 197 beat per minute but most of them did not reach their maximal heart rate and most of them tendency to choose number below than the highest RPE Brog CR-10.

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Increase in Heart rate also effect from loss of bodyweight due to dehydration and temperature also can affect on heart rate but in cool temperature our heart rate will slowly increase than in hot temoterature. When our body is at physical stress because of emotional or physical activity, our sympathetic nervous system will release adrenaline into bloodstream and effect of adrenaline to our body is increase in heart rate. When our brain send signal to our body that we need more energy to continue an activity while our body actually at it maximal effort and this make we to choose number below than the highest RPE Brog CR-10.
4.5.2 VO2max and RPE
Study for relationship between VO2max and RPE show linear significant different in post-hoc analyses in VO2 values (%VO2max) and RPE and small significant increase in trial 1 and trial 3 using the classic Brog scale in Roger G. Eston, et al., studies. Takeshi Ueda, et al., studies also show a significant between VO2 value (%VO2max) and RPE in pedaling and running but pedaling result show higher significant than running RPE (F = 15.012, p = However, in present finding show no significant between VO2max and RPE which using Brog CR10 scale. In James Faulkner, et al., studies has support this finding when the predicted VO2max and RPE show significant lower in all trials but between predicted and actual VO2max and RPE show no significant different when VO2 value were extrapolated to HR-max predicted.
Most of RPE been using are to compare test result with another test result or to compare it in group. Most of studies that compare VO2max with RPE show no significant result. It might be cause of subject’s that confuse to choose any number on scale cause not all number has word to indicated level of effort that need them to choose. In present study show that most of subjects choose number 8 and 9 to indicated their effort level. This because our brain still giving signal to our somatic nervous system(SNS) to stay longer on treadmill when our bodies are not able to fulfill the order make our decision to choose below than the highest number of Brog scale while our heart rate are at heart rate maximal.
4.5.3 VO2max and HR
Study for relationship between VO2max and HR show a correlation coefficient of 0.912 between increase in VO2max and HR in test 1 and test 2 by David V.B. James, et al. studies used only 8 male subjects which are runner athlete and average age of 25 ± 6 years old.
However, in present finding show that the correlation between VO2max and HR has no significant correlation result (r = .120, p = .605). Study by James Faulkner, ett al, showed there were no significant between actual and predicted VO2max when the VO2 value were extrapolated at HRmax predict p = >0.05. Jennifer C. R, et al study also support that VO2maxhas no significant result with HRmax p = >0.05.
A person with same age doing a same test will have different result of VO2max even through the HR are same it because other factor such as pulmonary system also play some role in affecting VO2max result. Total of oxygen being transport in blood through whole body can affect the VO2max result and intensity of training of a person also can cause an effect on VO2max. Increase in cardiac output and arterial-venous oxygen will increase VO2max because cardiac output is determined by HR and stroke volume.
4.5.4 VO2max and physical activity
Study for relationship between VO2max and physical activity showed that there is positive significant correlation result. These statements were supported by Sigurbjorn A. Arngrimsson, et al., where showed strong linear relationship between VO2max and physical activity which decrease in VO2max also decrease in physical performance for both gender. High relationship between VO2max and physical activity also was support by James Faulkner and Roger Eston, which show VO2max in high fitness for men are better than VO2max for men that have low fitness level.
However, in present finding showed that the correlation between VO2max and physical activity are not strong as previous studies. James Faulkner, et al., showed both high fitness and low fitness in men has no significant different in average correlation using Fisher’s Z transformation procedure.
A person that has physical activity more than 6 hour in a week can’t be said will have greater VO2max than a person has less than 6 hour of physical activity, because it depend on what type of physical activity of person done in a week. A person that spend more than 6 hour a week in weight training program will have different VO2max compare to person training less than 6 hour a week on cardio training program. This is a reason we can’t assume a person that having physical activity less than 6 hour a week will have low score in VO2max than a person having more than6 hour a week on physical activity.
 

Correlation Between Heart Rate And Vo2 Physical Education Essay

The experiments aims were to run subjects to exhaustion on an increasing work rate in order to calculate changes in heart rate and VO2 consumption, in order to see if there was a correlation. Work rate started at 8km/h and increased by 2 km/h every 3 minutes. A strong positive correlation between increased work rate and a proportional increase in both heart rate and VO2 was found when assessing the results and formulating the graphs. All subject were male: age (19.43years ± 0.79), height (1.82 m ±0.10), weight (73.29kg ±15.11).
The aim was to determine oxygen uptake and carbon dioxide production at rest and during a bout of sub maximal exercise and what relationship it had with heart rate.
Heart rate itself is simply “the amount of work the heart must do to meet the increased demands of the body when engaging in activity” Wilmore and Costill 2004 page 224) measure it is usually done at the radial of carotid artery sites.
Heart rate in this case is an indicator of the relative stress placed on the cardio respiratory system during movement. “Heart rate increases during physical activity are controlled by the synoatorial node, in response to decreased parasympathetic neural stimulation, increased sympathetic neural stimulation- a phenomenon termed cardiac chronotropic regulation” (Robberts 1996, Page 144). Thus clearly an increase in heart rate when exercising will lead to an increase in cardio output, this is supported by Wilmore and Costill “when you exercise, your heart rate increases directly in proportion to the exercise intensity”. (Wilmore and Costill 2004 page 224) With regards to this experiment and the increase in work rate and the gradual increase in heart rate observed this is backed up by (Freedson and Miller 2000; Robergs 1996) “moderate through vigorous heart rate increases linearly and proportionally with the intensity of movement and the volume of oxygen consumed.” An important factor relating to the first heart rate and VO2 taken is that other factors may effect the subjects, for example stress, fluid levels and digestion, digestion due to parasympathetic. VO2 is the volume of oxygen consumed per minute. Recent research “has shown that changes in plasma volume are highly correlated with changes in stroke volume and VO2 making the training-induced increases in plasma volume one of the most significant training effects” (Wilmore and Costill 2004 page 287). With regards to the respiratory system the reason its function is not limited is due to a greater ventilation capacity than cardiovascular function.

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The differences in Pulmonary Ventilation is vast. “in untrained subjects maximal pulmonary ventilation typically increases from a beginning rate of about 100 to 120 L/min to about 130 to 150 L/min or more following training” (Wilmore and Costill 2004 page 287). However in extremely well trained athletes it may exceed 240L/min. The reasons for this vast change are “increased tidal volume and increased respiratory rate at maximal exercise” (Wilmore and Costill 2004 page 287). The inspiratory muscles are also important as training these can help with better performance. Dispite this being the case Wilmore and Costill (2004)conclude that “in highly trained person’s adaptation the pulmonary systems capacity for oxygen transport won’t be able to meet the demands of the limbs and the cardiovascular system” (Wilmore and Costill 2004 page 287). In other words even if you are unbelievably fit there will still be physical limitations. When this takes place in this case it effects ‘arterial oxygen saturation’ in fact it gets lower “decreases by 96%” (Wilmore and Costill 2004 page 287).
A factor that will have effects all subjects is their lactate threshold. In a trained athlete lactate will still be produced but at a later stage than id they were untrained and the level of lactic acid will often be lower as the body is able to deal with it better.
The latter stages of the experiment were seen to be more important as it is where breathing is intensified and a higher oxygen consumption occurs, a theory backed by Adams 2002 pg 186, “latter stages of the exercise protocol is when maximal oxygen consumption occurs.”
Method
The experiment was approved by the Durham ethics committee and all 7 male subjects signed a consent form. All subjects in the experiment are young ranging from 18 to 21 and thus age is not a deciding factor in the maximal heart rates.
Table 1
 
Age
Height
Mass
Mean
19.43
1.82
73.29
St Dev
0.79
0.10
15.11
Resting heart rate from a polar N23 heart rate monitor and VO2 from a Harvard Douglas bag were recorded as well as weight and age. Subject were put on a hp cosmos Germany 401 treadmill on a set gradient of… subject steps on to the moving treadmill and the stop watch was started. In the last minute of every 3 minutes the mouth piece was attached to the subject and oxygen excretion for the minute was collected. The subject shouted out the intensity on a Borgs 6-20 RPE scale. At the end of every 3 minutes the intensity was increased. This protocol was continued until exhaustion. The exact time of stopping was then recorded.
Four minutes after exhaustion blood is taken in order to calculate the subjects lactic acid levels. 1 litre of expired air was then removed in a small Douglas bag and put into the Servomex 1440 Cranley medical uk gas analyser for levels of oxygen and carbon dioxide. Original Douglas bag is attached to a dry gas analyser and measures the volume of expired air. The 1 litre earlier removed is then added to this total.
Results
Figure 1: A bar chart displaying means and standard deviations of the relationship between Heart rate and VO2.
Figure 2: Bar chart displaying means and standard deviations of the relationship between work rate and heart rate.
Figure 3: Bar chart displaying mean and standard deviations of the relationship between VO2 (ml.kg.min) and work rate.
Discussion
It is clear that the data’s showing a strong positive correlation as hypothesised in Fig 1. An increase in work rate means an increase in both heart rate and VO2, “When exercise is performed at a given work rate which is below lactate threshold (LT), Oxygen uptake increases exponentially to a steady-state level”( Xu F; Rhodes E.C. 1999).Fig 2 which is showing the relationship between work rate and heart rate is precise in displaying the increase in heart rate directly to work rate, it should be noted that “When exercise is performed at a work rate above LT, the Oxygen uptake kinetics become more complex. An additional component is developed after a few minutes of exercise. The slow component either delays the attainment of the steady-state Oxygen uptake or drives the Oxygen uptake to the maximum level, depending on exercise intensity” ( Xu F; Rhodes E.C. 1999).
Fig 3is displaying mean and standard deviations of the relationship between VO2 (ml.kg.min), and work rate, it is to be expected that if heart rate increases due to work rate then VO2 will also do the same as if it didn’t then the heart would not be able to either.
Despite this there are still contradicting theories, Adams also concludes that there are other factors that need to be accounted for, “other factors such as running economy and the ability to use a high percent of the maximal oxygen consumption (fractional utilization) and ventilator threshold also the important indicators of success in aerobic performance.” (Adams page 202). An expectation of further studies with fractional utilization of VO2 and time as well as ventilator threshold. It must also be noted that oxygen consumption is dependent on sex and age, “maximal oxygen consumption is higher in men than women, and in younger adults than in older adults.” (Adams pg 202).
Xu and Rhodes believe there is a slow component during intensive exercise, they go on further and speculate some of the possible causes, “The possible causes for the slow component of Oxygen uptake during heavy exercise include: increases in blood lactate levels, increases in plasma epinephrine (adrenaline) levels, increased ventilatory work, elevation of body temperature and recruitment of type IIb fibres” ( Xu F; Rhodes E.C. 1999). These are clearly all contributing factors however it is the recruitment of IIb fibers that Xu and Rhodes attribute most to an incrased need for oxygen, “During high intensity exercise an increase in the recruitment of low-efficiency type IIb fibres can cause an increase in the oxygen cost of exercise. A change in the pattern of motor unit recruitment, and thus less activation of type IIb fibres, may also account for a large part of the reduction in the slow component of Oxygen uptake observed after physical training.” ( Xu F; Rhodes E.C. 1999). I would expect more research focused on this factor in order to examine why there is a need for such a high percentage “86%”( Xu F; Rhodes E.C. 1999) of oxygen uptake slow component obtained from the limbs and more particularly IIb fibers.
 

Correlation and Simple Linear Regression

 Correlation and simple linear regression methods assess the degree of strength, direction of association, and a linear summary of relationship existing between two variables, or observational units (Berg, 2004). In an effort to expose the descriptive analysis, correlational patterns resulting from the dataset DEL618_DHS618m1.sav, the writer/researcher hopes to examine the associative factors in light of the inferential statistics procedures that are paramount to the assignment. Such endeavor should help the writer/researcher to meet the goal of the theoretical basis for the assignment.

Correlation and Linear Frameworks

The correlation and linear patterns usually found in statistical analyses indicate that the role of independent and dependent variables is essential in the analysis of data as well as the levels of measurement utilized. In bivariate statistics and regression, as Berg (2004), and Myers, Gamst, and Guarino (2006) asserted, a flexibility of roles of the variables: playing one role in one context, and another role in another context can help explain their effects based on data collection methods used. This is important for the type of research design, the writer/researcher posits.

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A linear regression shows how a distribution is presented depending on the values of a variable x, and how another variable y varies. The relationship between these variables is the key concern. There is an effort to define a best line to ascertain the paths of the measures of central tendency (mean, variance, standard deviation…) (Berg, 2004, p.24). A simple linear equation is defined as y = a+ bx, where y is defined as the dependent variable, and x as the independent variable. The intercept, a constant, is labeled as a, and b the coefficient, is also considered as a slope. A bivariate relationship captured in a scatterplot shows how the relationship, and the shape of the bivariate between the variables are presented.

Statistical Basis

The focus of this assignment is generated from a researcher’s willingness to examine factors influencing reading scores among school children. 7 variables considered yield substantive descriptive statistics showing whether correlation relationships exist among the variables. The descriptive statistics in tables 1, 2, 3 and 4 SPSS below provide a complete picture of the variables, frequencies generated, and guided the writer/researcher description of the results. The descriptive statistics show that females have a higher frequency percentage (55.6 %) than males (24.4%). Reading rank has a higher mean than visual acuity, a lower standard deviation, and variances compared to visual acuity. This seems to suggest that the predictor and outcome variables can be considered in a bivariate domain, and correlation designs, as one variable relate to another; but there are other missing factors and further analysis to substantiate a valid conclusion or result (Keppel, Saufley, & Tokunaga, 1992).

The mean, standard deviation, median, and variances for reading rank are: 16.43 (mean), standard deviation (6.188), median (18), and variance (38.287), compared to visual acuity of  10.41 (mean), standard deviation (6.254), median (10.00) median, and 39.114(variances). For every standard deviation increase in visual acuity, there will be a 6.188 standard deviation in reading rank values.

Furthermore, for visual acuity rank, with a standard deviation of SD =6.254, skewness of .121, as depicted in table 5, and reading rank with SD =6.188, and skewness of -.965, suggesting both a skewness
Descriptive Statistics

Table 1. Descriptive Statistics. Selected Variables

Variables

Group

Mean

Standard Deviation

Median

 Frequencies (%)

Variances

Gender

1.56

.502

2.00

.252

Female

30 (55.6)

Male

24 (44.4)

Total

54 (100)

ID

12.94

6.534

13.00

42.645

1

1(1.9)

2

1 (1.9)

3

1 (1.9)

4

4 (7.4)

5

1 (1.9)

6

1 (1.9)

7

4 (7.4)

8

4 (7.4)

9

4 (7.4)

10

1 (1.9)

11

1 (1.9)

12

14 (1.9)

13

4 (7.4)

14

4 (7.4)

15

4 (7.4)

16

1 (1.9)

17

1 (1.9)

18

1 (1.9)

19

4 (7.4)

20

4 (7.4)

21

1 (1.9)

22

1 (1.9)

23

1 (1.9)

24

4 (7.4)

Total

54 (100)

Reading in Rank

16.43

6.188

18.00

38.287

1

1 (1.9)

2

1 (1.9)

3

1 (1.9)

4

1 (1.9)

5

1 (1.9)

6

1 (1.9)

7

1 (1.9)

8

1 (1.9)

9

1 (1.9)

10

1 (1.9)

11

1 (1.9)

12

2 (3.7)

13

1 (1.9)

14

1 (1.9)

15

4 (7.4)

16

4 (7.4)

17

4 (7.4)

18

4 (7.4)

19

4 (7.4)

20

4 (7.4)

21

4 (7.4)

22

4 (7.4)

23

4 (7.4)

24

4 (7.4)

Total

54 (100)

Visual Acuity in Rank

10.41

6.254

10.00

39.114

1

4 (7.4)

2

4 (7.4)

3

1 (1.9)

4

1 (1.9)

5

4 (7.4)

6

4 (7.4)

7

4 (7.4)

8

1 (1.9)

9

1 (1.9)

10

5 (9.3)

11

4 (7.4)

12

1 (1.9)

13

1 (1.9)

14

1 (1.9)

15

5 (9.3)

16

1 (1.9)

17

1 (1.9)

18

1 (1.9)

19

5 (9.3)

20

5 (9.3)

Total

54 (100)

 

Table 2 Descriptive Statistics

N

Minimum

Maximum

Mean

Std. Deviation

Variance

Gender

54

1

2

1.56

.502

.252

Visual Acuity Rank

54

1

20

10.41

6.254

39.114

ReadingRank

54

1

24

16.43

6.188

38.287

ID

54

1

24

12.94

6.534

42.695

Valid N (listwise)

54

Table 3.Visual Acuity Rank Frequencies

Frequency

Percent

Valid Percent

Cumulative Percent

Valid

1

4

7.4

7.4

7.4

2

4

7.4

7.4

14.8

3

1

1.9

1.9

16.7

4

1

1.9

1.9

18.5

5

4

7.4

7.4

25.9

6

4

7.4

7.4

33.3

7

4

7.4

7.4

40.7

8

1

1.9

1.9

42.6

9

1

1.9

1.9

44.4

10

5

9.3

9.3

53.7

11

4

7.4

7.4

61.1

12

1

1.9

1.9

63.0

13

1

1.9

1.9

64.8

14

1

1.9

1.9

66.7

15

5

9.3

9.3

75.9

16

1

1.9

1.9

77.8

17

1

1.9

1.9

79.6

18

1

1.9

1.9

81.5

19

5

9.3

9.3

90.7

20

5

9.3

9.3

100.0

Total

54

100.0

100.0

Table 4. Reading Rank Frequencies

Frequency

Percent

Valid Percent

Cumulative Percent

Valid

1

1

1.9

1.9

1.9

2

1

1.9

1.9

3.7

3

1

1.9

1.9

5.6

4

1

1.9

1.9

7.4

5

1

1.9

1.9

9.3

6

1

1.9

1.9

11.1

7

1

1.9

1.9

13.0

8

1

1.9

1.9

14.8

9

1

1.9

1.9

16.7

11

1

1.9

1.9

18.5

12

2

3.7

3.7

22.2

13

1

1.9

1.9

24.1

14

1

1.9

1.9

25.9

15

4

7.4

7.4

33.3

16

4

7.4

7.4

40.7

17

4

7.4

7.4

48.1

18

4

7.4

7.4

55.6

19

4

7.4

7.4

63.0

20

4

7.4

7.4

70.4

21

4

7.4

7.4

77.8

22

4

7.4

7.4

85.2

23

4

7.4

7.4

92.6

24

4

7.4

7.4

100.0

Total

54

100.0

100.0

Table 5. Descriptive Statistics and Skewness

N

Minimum

Maximum

Mean

Std. Deviation

Variance

Skewness

Statistic

Statistic

Statistic

Statistic

Statistic

Statistic

Statistic

Std. Error

Gender

54

1

2

1.56

.502

.252

-.230

.325

Visual Acuity Rank

54

1

20

10.41

6.254

39.114

.121

.325

Reading Rank

54

1

24

16.43

6.188

38.287

-.965

.325

ID

54

1

24

12.94

6.534

42.695

.066

.325

Valid N (listwise)

54

Figure 1. Scatterplot for visual Acuity and Reading Rank Matrix

Bivariate Statistic and Regression

 

Multiple Regression and P-values: a) Relationship Between Social Studies, Math,

and Reading. To ascertain the relationship between these variables, the writer/researcher

posits the following research questions, and hypotheses:

RQ1: What is the relationship between social studies, math, and reading considered concurrently?

Ho: There is no statistically significant relationship between social studies, math, and reading considered concurrently.

Halt: There is a statistically significant relationship between social studies, math, and reading considered concurrently.

SPSS correlations in table 1 results reveal a correlation of .342 between social studies and math, and .647 between social studies and reading. The analysis revealed a significant and positive correlation between math and reading. r=. 342, and p= .011 for math; and r=.647, with p = .000, p

Table 1. Correlations

Social Studies

Math

Reading

Social Studies

Pearson Correlation

1

.342*

.647**

Sig. (2-tailed)

.011

.000

N

54

54

54

Math

Pearson Correlation

.342*

1

.423**

Sig. (2-tailed)

.011

.001

N

54

54

54

Reading

Pearson Correlation

.647**

.423**

1

Sig. (2-tailed)

.000

.001

N

54

54

54

*. Correlation is significant at the 0.05 level (2-tailed).

**. Correlation is significant at the 0.01 level (2-tailed).

b) Partial correlation coefficients for variables reading and social studies with gender held as a constant.

RQ2: What is the relationship between reading and social studies with gender held constant?

Ho: There is no statisticallysignificant relationship between reading and social studies with gender held constant.

Halt: There is a statisticallysignificant relationship between reading and social studies with gender held constant.

A partial coefficient using bivariate regression analysis was conducted for the variable reading and social studies, holding the variable gender as constant. The analysis shows a positive correlation between social studies and reading r = .643, p= df (51), and p value

Table 2 Partial Correlation Reading and Social Studies

Control Variables

Reading

Social Studies

Gender

Reading

Correlation

1.000

.643

Significance (2-tailed)

.

.000

df

0

51

Social Studies

Correlation

.643

1.000

Significance (2-tailed)

.000

.

df

51

0

c) Partial correlation coefficients for variables math and reading with gender held as a constant.

RQ3: Is there a relationship between math and reading with gender being held constant?

Ho: There is no statisticallysignificant relationship between math and reading with gender being held constant.

Ho: There is no statisticallysignificant relationship between math and reading with gender being held constant.

A SPSS analysis was conducted for the variables math and reading with gender held as a constant. The analysis showed a positive correlation with r =. 354, df (51), the correlation between math and reading was significant.p= .009

 

     Table 3. Partial Correlation with Math and Reading

Control Variables

Math

Reading

Gender

Math

Correlation

1.000

.354

Significance (2-tailed)

.

.009

df

0

51

Reading

Correlation

.354

1.000

Significance (2-tailed)

.009

.

df

51

0

Appropriate Test to Measure Correlation Between Reading in Rank, and Visual

Acuity:  A bivariatetest would be appropriate to measure the correlation between reading in rank, and visual acuity. SPSS analysis revealed a negative correlation between reading rank and visual acuity. r = -0.067, p =0.628

Table 4. Reading in Rank and Visual Acuity Correlations

ReadingRank

Visual Acuity Rank

ReadingRank

Pearson Correlation

1

-.067

Sig. (2-tailed)

.628

N

54

54

Visual Acuity Rank

Pearson Correlation

-.067

1

Sig. (2-tailed)

.628

N

54

54

 

 

Linear Regressions

1) Social Studies as a Predictor of Reading Scores: Consistent with the notion of 

interactive effects of variables, espoused by Agresti (2011), linear regressions

summarizing relationships between variables ought to make sense of data, and seem to suggest more than one predictor is needed for generalizing regression analyses (Berk, 2004, p.21).

A bivariate regression analysis was conducted for social studies as a predictor for social studies. R2 =.419, F (1, 52) =37.449, reading = 55.347, Social studies = 4.257,

and p

Table 5  Variables Entered/Removeda

Model

Variables Entered

Variables Removed

Method

1

Social Studiesb

.

Enter

a. Dependent Variable: Reading

b. All requested variables entered.

 

Table 6. Model Summary

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

Change Statistics

R Square Change

F Change

df1

df2

Sig. F Change

1

.647a

.419

.407

11.113

.419

37.449

1

52

.000

a. Predictors: (Constant), Social Studies

 

Table 7ANOVAa

Model

Sum of Squares

df

Mean Square

F

Sig.

1

Regression

4624.798

1

4624.798

37.449

.000b

Residual

6421.739

52

123.495

Total

11046.537

53

a. Dependent Variable: Reading

b. Predictors: (Constant), Social Studies

Table 8 Coefficientsa

 

 

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

95.0% Confidence Interval for B

Collinearity Statistics

B

Std. Error

Beta

Lower Bound

Upper Bound

Tolerance

VIF

1

(Constant)

55.347

7.597

7.285

.000

40.102

70.591

Social Studies

4.257

.696

.647

6.120

.000

2.861

5.652

1.000

1.000

a. Dependent Variable: Reading

Table 9 Bivariate Regression for Social Studies as Predictor of Reading

 

Regression Weights

Variables                                            b                   B

Social studies                                   4.257           .647

R2                         .419  

F              37.449         

 

2) Math as a Predictor of Reading Scores: A bivariate regression was conducted for math and reading scores. Tables 11, 12, and 13 summarized analysis results. The bivariate regressions regression model with social studies as a predictor shows:

R2= .179

R2= .163 (adjusted), F (1, 52) =11.326 is significantly significant p=.001, and reading=74.549+1.155(math).

Table 11 below shows, B=.423 for math; thus for every standard deviation increase in math scores there is a .423 standard deviation rise in reading scores.

Table 10. Bivariate Regression for Math as a Predictor of Reading Scores

                                             Regression weighs

Variables                                 b                 B

Math                                   1.155            .423___________________________

Table 11 Model Summary

Model

R

R Square

Adjusted R Square

Std. Error of the Estimate

Change Statistics

R Square Change

F Change

df1

df2

Sig. F Change

1

.423a

.179

.163

13.208

.179

11.326

1

52

.001

a. Predictors: (Constant), Math

Table 12 ANOVAa

 

Model

Sum of Squares

df

Mean Square

F

Sig.

1

Regression

1975.717

1

1975.717

11.326

.001b

Residual

9070.820

52

174.439

Total

11046.537

53

a. Dependent Variable: Reading

b. Predictors: (Constant), Math

 

Table 13 Coefficientsa

Model

Unstandardized Coefficients

Standardized Coefficients

t

Sig.

95.0% Confidence Interval for B

Collinearity Statistics

B

Std. Error

Beta

Lower Bound

Upper Bound

Tolerance

VIF

1

(Constant)

74.549

8.036

9.277

.000

58.424

90.674

Math

1.155

.343

.423

3.365

.001

.466

1.844

1.000

1.000

a. Dependent Variable: Reading

The SPSS outputs are included for further review by the professor if needed.

References

Agresti, A. (2011). An introduction to categorical data analysis (2nd Ed). Hoboken, NJ: John Wiley & Sons.

Berk, R.A. (2004). Regression analysis. A constructive critique. Thousand Oaks, CA: Sage.

Keppel, G., Saufley, W.H., Jr., & Tokunaga, H. (1992). Introduction to design and analysis: A student’s handbook (2nd ed.). New York: W.H. Freeman.

Meyers, L. S., Gamst, G., & Guarino, A.J. (2006). Applied multivariate research. Design and interpretation. Thousand Oaks, CA: Sage.

 

 
 

Correlation the Number of the Students’ College Applications and Consumption of Caffeine

 

Correlation the Number of the Students’ College Applications and Consumption of Caffeine

Introduction and statement of intent:

Students are exposed to a lot of stress during the college admissions process. These increased stress levels come with some negative consequences like the increase in consumption of caffeine. The objective of this academic project is to find out whether the number of colleges high school seniors are applying to increases the consumption of stimulant substances like caffeine. I will explore the relationship between the amount of college applications and the daily consumption of coffee within the population of high school seniors. I selected this topic because, as a senior, the massive number of college applications need to submit has affected noticeably the quantity of caffeine I personally consume. On the basis of this study I will try to know if this is a rule that can be applied in a more general basis or not.

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In order to collect the primary data, I used direct interview method. For accurate and reliable information, the data was collected from 50 seniors in different schools in different countries. I collected information regarding different possible variables in order to be able to decide which one would give me the most accurate information for the exploration. Finally, I decided to stick to the data related to the number of cups of coffee that a student takes daily and the number of colleges that student is applying to. I will be using in order to get to a conclusion, my data, several mathematical calculations to see if there is a relation between the two variables and use of scatter diagram for better presentation and understanding of the topic.

Initially, to find out about the relationship between my two variables (number of college applications and coffee consumption), my data will be plotted on an scatter-diagram, as well as find the Pearson’s correlation coefficient (r). I will effectuate a rather a chi-squared test or the regression equation depending on the value of r that obtain to make some predictions. This, will allow me to determine the liability of the regression line to make predictions about my data, and if the variables are independent or not by testing the null hypothesis in the chi-squared test if necessary.

Correlation Analysis:

Correlation is used to define the linear relationship between two continuous variables. A correlation could be positive, (both variables move in the same direction), or negative (when one variable’s value increases, the other variables’ values decrease). Correlation can also be zero (variables are unrelated). The strength/qualitatively and direction of the linear relationship between two or more variables.

Correlation allows me to clearly and easily see if there is a relationship between variables. This relationship can then be displayed in a graphical form. It also gives a precise quantitative value indicating the degree of relationship existing between the two variables and it measures the direction and the relationship between the two variables.

Assumptions:

There are some vital assumptions are required to establish correlation between two variables. These assumptions are:

Both variables (Number of Applications and Number of Cups of Coffee a day (8 oz.)) is normally distributed.

Second assumption includes linearity. Linearity assumes that there is a straight-line relationship

between each of the two variables and on other hand.

Mathematical investigation:

Data:

 

Number of

applications

Number of cups

of coffee a day (8 oz.)

 

Number of applications

Number of cups

of coffee a day (8 oz.)

1

10

4

26

11

3

2

15

5

27

6

2

3

3

0

28

16

7

4

11

6

29

20

8

5

11

5

30

11

3

6

19

9

31

18

5

7

6

2

32

9

3

8

12

4

33

16

3

9

5

1

34

2

0

10

17

7

35

10

3

11

21

8

36

9

2

12

10

1

37

6

1

13

8

3

38

4

0

14

6

2

39

13

2

15

7

2

40

21

7

16

10

0

41

14

3

17

14

4

42

2

1

18

13

3

43

10

3

19

1

0

44

7

2

20

12

3

45

15

6

21

15

6

46

4

1

22

18

9

47

12

4

23

4

1

48

1

0

24

7

2

49

15

6

25

24

8

50

11

2

 

Mathematical Process to get the Correlation Coefficient (r):

Step 1: Making a chart with data for two variables, labelling the variables (x) and (y), and add three more columns labelled (XY), (X2), and (Y2).

Step 2: Complete the chart using basic multiplication of the variable values.

Computation of Karl Pearson’s Coefficient of correlation: (Direct Method)

 

Number of Applications

Number of Cups of Coffee a day (8 oz.)

XY

X2

Y2

1

10

4

40

100

16

2

15

5

75

225

25

3

3

0

0

9

0

4

11

6

66

121

36

5

11

5

55

121

25

6

19

9

171

361

81

7

6

2

12

36

4

8

12

4

48

144

16

9

5

1

5

25

1

10

17

7

119

289

49

11

21

8

168

441

64

12

10

1

10

100

1

13

8

3

24

64

9

14

6

2

12

36

4

15

7

2

14

49

4

16

10

0

0

100

0

17

14

4

56

196

16

18

13

3

39

169

9

19

1

0

0

1

0

20

12

3

36

144

9

21

15

6

90

225

36

22

18

9

162

324

81

23

4

1

4

16

1

24

7

2

14

49

4

25

24

8

192

576

64

26

11

3

33

121

9

27

6

2

12

36

4

28

16

7

112

256

49

29

20

8

160

400

64

30

11

3

33

121

9

31

18

5

90

324

25

32

9

3

27

81

9

33

16

3

48

256

9

34

2

0

0

4

0

35

10

3

30

100

9

36

9

2

18

81

4

37

6

1

6

36

1

38

4

0

0

16

0

39

13

2

26

169

4

40

21

7

147

441

49

41

14

3

42

196

9

42

2

1

2

4

1

43

10

3

30

100

9

44

7

2

14

49

4

45

15

6

90

225

36

46

4

1

4

16

1

47

12

4

48

144

16

48

1

0

0

1

0

49

15

6

90

225

36

50

11

2

22

121

4

 

542

172

2496

7444

916

 

Step 3: After multiplied all the values to complete the chart, add up all of the columns from top to bottom.

Step 4: Use the following formula to find the Pearson correlation coefficient value.

Pearson’s Correlation Coefficient (r):
r=n xy–(x*y)nx2–(x)2* ny2–(y)2 
r=502,496–(542*172)50 7,444–(542)2* 50(916)–(172)2
r=1,24,800–93,2243,72,200– 2,93,764* 45,800–29,584
r=31,57678,436* 16,216
r=31,576280.06427* 127.3420 r=31,57635,663.9442 =0.89

 

Step: 6 Solving with the formula above by plugging in all the correct values, the result is thecoefficient value.

If the value is a negative number, then there is a negative correlation, and if the value is a positive number, then there is a positive correlation.

Pearson’s Correlation coefficient

Sr. No.

X

Y

(X–X̅)

(X–X)2̅

(Y–Y̅)

(Y–Y)2̅

X–X̅(Y–Y̅)

1

10

4

-0.84

0.71

0.56

0.31

-0.47

2

15

5

4.16

17.31

1.56

2.43

6.49

3

3

0

-7.84

61.47

-3.4

11.83

26.97

4

11

6

0.16

0.03

2.56

6.55

0.41

5

11

5

0.16

0.03

1.56

2.43

0.25

6

19

9

8.16

66.59

5.56

30.91

45.37

7

6

2

-4.84

23.43

-1.4

2.07

6.97

8

12

4

1.16

1.35

0.56

0.31

0.65

9

5

1

-5.84

34.11

-2.4

5.95

14.25

10

17

7

6.16

37.95

3.56

12.67

21.93

11

21

8

10.16

103.23

4.56

20.79

46.33

12

10

1

-0.84

0.71

-2.4

5.95

2.05

13

8

3

-2.84

8.07

-0.4

0.19

1.25

14

6

2

-4.84

23.43

-1.4

2.07

6.97

15

7

2

-3.84

14.75

-1.4

2.07

5.53

16

10

0

-0.84

0.71

-3.4

11.83

2.89

17

14

4

3.16

9.99

0.56

0.31

1.77

18

13

3

2.16

4.67

-0.4

0.19

-0.95

19

1

0

-9.84

96.83

-3.4

11.83

33.85

20

12

3

1.16

1.35

-0.4

0.19

-0.51

21

15

6

4.16

17.31

2.56

6.55

10.65

22

18

9

7.16

51.27

5.56

30.91

39.81

23

4

1

-6.84

46.79

-2.4

5.95

16.69

24

7

2

-3.84

14.75

-1.4

2.07

5.53

25

24

8

13.16

173.19

4.56

20.79

60.01

26

11

3

0.16

0.03

-0.4

0.19

-0.07

27

6

2

-4.84

23.43

-1.4

2.07

6.97

28

16

7

5.16

26.63

3.56

12.67

18.37

29

20

8

9.16

83.91

4.56

20.79

41.77

30

11

3

0.16

0.03

-0.4

0.19

-0.07

31

18

5

7.16

51.27

1.56

2.43

11.17

32

9

3

-1.84

3.39

-0.4

0.19

0.81

33

16

3

5.16

26.63

-0.4

0.19

-2.27

34

2

0

-8.84

78.15

-3.4

11.83

30.41

35

10

3

-0.84

0.71

-0.4

0.19

0.37

36

9

2

-1.84

3.39

-1.4

2.07

2.65

37

6

1

-4.84

23.43

-2.4

5.95

11.81

38

4

0

-6.84

46.79

-3.4

11.83

23.53

39

13

2

2.16

4.67

-1.4

2.07

-3.11

40

21

7

10.16

103.23

3.56

12.67

36.17

41

14

3

3.16

9.99

-0.4

0.19

-1.39

42

2

1

-8.84

78.15

-2.4

5.95

21.57

43

10

3

-0.84

0.71

-0.4

0.19

0.37

44

7

2

-3.84

14.75

-1.4

2.07

5.53

45

15

6

4.16

17.31

2.56

6.55

10.65

46

4

1

-6.84

46.79

-2.4

5.95

16.69

47

12

4

1.16

1.35

0.56

0.31

0.65

48

1

0

-9.84

96.83

-3.4

11.83

33.85

49

15

6

4.16

17.31

2.56

6.55

10.65

50

11

2

0.16

0.03

-1.4

2.07

-0.23

 

542

172

 

1568.72

 

324.32

631.52

 

Math Processes:

The Pearson’s correlation coefficient is calculated as the covariance of the two variables divided by the product of the standard deviation of each data sample. It is the normalization of the covariance between the two variables to give an interpretable result.

The use of mean and standard deviation in the calculation suggests the need for the two data samples to have a normal distribution.

In the result of the calculation, the correlation coefficient can be interpreted to understand the relationship.

Pearson’s Coefficient of Correlation: (r)

X̅= xin

X̅= 54250 = 10.84

 

Y̅= yin

Y̅= 17250=3.44

 

Sx=(X–X)2̅n

Sx=1,568.7250= 5.60

 

Sy=(Y–Y)2̅n

Sy=324.3250 =2.55

 

Sxy= X–X̅(Y–Y)̅n

Sxy= 631.52 50

 

Sxy=12.63

 

Interpretation or results:

The correlation coefficient is a measure of the strength of the linear trend relative to the variability of the data around that trend. Thus, it is dependent both on the magnitude of the trend and the magnitude of the variability in the data.

There is high-degree of positive correlation between Number of applications and Number of cups of coffee a day (8 oz.). This implies that if there is increase in number of applications there is also increase in number of cups of coffee. Conversely, if there is decrease in number of applications there is also decrease in number of cups of coffee.

r=SxySx*Sy

r=12.635.60*2,55

 r=0.89

Scatter Diagram of Correlation:

Scatter Diagram:

 

 

 

In the scatter diagram it is shown, the value of the independent variable (number of applications) along the X-axis and the value of the dependent (number of cups of coffee a day) on the Y-axis. For each pair of X and Y values (Xi, Yi), I plotted dots on graph paper for the pairs of observation. The diagram of dots obtained is called scatter diagram.

From the scatter diagram we can know the direction of correlation (positive or negative) but we cannot know the degree of correlation (numerical value of correlation) between the two variables. By looking to the scatter of the various points, we can get an idea whether the variables are related or not.

Discussion/ Validity:

The Pearson correlation coefficient measures the strength of the linear relation between two variables. It can be used to estimate the population correlation, ρ.

This graph shows that there is highly positive correlation between number of applications and number of cups of coffee a day (8 oz.) among high school students.

Limitations:

There are some limitations of this research project which may lead to wrong conclusions:

Correlation are always based on available data and it does not allow me to go beyond this data. There are number of factors other than number of applications and (Independent Variable) which may influence on number of cups of coffee a day (8 oz.) (Dependent Variable) of the students– such as stress, socio economic conditions, relation with friends and family members etc.

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To establish relation between two variables it is assumed that there is linear relationship between them, whether such kind of relationship may exist or not. Some of the observations affect the value of coefficient of correlation and may provide misleading information. Finally, if there is a strong correlation between two variables it does not imply there is a cause and effect relationship between them. In this case, other factors can influence this relationship.

Conclusion:

On the basis of the statistical data and mathematical calculations, there is a high degree of positive correlation between number of applications and number of cups of coffee a day (8 oz.). The number of college students who suffer from stress-related problems appears to be on the rise.

Occasional stress is an unavoidable part of our routine life. Small amounts of stress can even have a positive effect on students, allowing us to push themselves when they encountered a difficult task. However, high levels of stress on students over a prolonged period of time are linked to increased rates of depression, anxiety, cardiovascular disease, and other potentially health-threatening issues. This conclusion shows us how important it is to learn how to manage stress by students before they suffer any adverse effects. It may be useful to potentiate the rise of awareness in identification of potential stress risks, stress management techniques, and resources that should be available to all high school students. However, we should not forget there are other factors that may affect the relationship between these two variables.

Bibliography:

Correlation. (n.d.). Retrieved January 5, 2019, from http://www.stat.yale.edu/Courses/1997-98/101/correl.htm

Correlation of Mathematics with Other Subjects

“No subject is ever well understood and no art is intelligently practiced, if the light which the other studies are able to throw upon it is deliberately shut out.” – RAMONT

What is correlation?
The meaning of term ‘correlation’ in simplest form is “connect” or “to be connected”. More precisely, ‘Correlation’ means mutual relation of two or more items/things or Mutual relation of two or more than two items/things.
The relation may be inverse or direct. For examle if there are two variables ‘a’ and ‘b’, if there is increase or decrease in one will affect on other. It is really a brainstorming activity which involves lots of efforts to be establishing relationship between school Subjects.
In modern scenario, due to the number of innovation of 21st century involves lots of development in the education. These involve meaningful knowledge of the concept. Here child’s mind is an integrated whole, he wants to receive experiences in a fruitful manner.
Definition
“Correlation indicates a joint-relationship between two variables.” Lathrop
“Correlation is concerned with describing the degree of relation between variables.” Ferguson
“Correlation is an analysis of co-variation between two or more variables.” A.M. Taule
“Correlation analysis deals with the association between two or more variables.” Simpson & Kafka
Types of correlation

Positive correlation – when an increase in one variable, increases the value of another variable or vice versa.
Negative correlation – when an increase in one variable, decreases the value of another variable or vice versa.
Incidental correlation — It is not a planned/pre-decided, no deliberate or a systematic attempt made to correlate it. Teacher plays a leading role. E.g. If any teacher has basic knowledge of concepts/elements of versatile or different subjects, he can easily make correlation between two or more subjects. A teacher cannot establish incidental correlation without having knowledge of different subjects.
Systematic correlation – teachers can sit together with his students and how to correlate? While doing systematic correlation, the previous knowledge/content of the student should be related with the current knowledge. To relate the same is called systematic correlation. Here, the student and the teacher have to think about the application of the fact, laws, principles, and correlation of two subjects. After that knowledge becomes interesting.
No correlation – when there is no mutual relationship between the two variables. It is also known as no linear dependent.

Uses of correlation:

The aim of education is “to achieve the all round development of a child”, this cannot be done by teaching only in simple classroom.
In correlation, the practical subjects like maths and science plays important role. Where the correlation with concepts is used in learning of students.
It makes learning permanent and concrete and knowledge to the learner.
It makes the lesson easy and clear for the student.
It enhances the mental abilities like problem solving, logical reasoning, imagination, and analytical power of student, because these can easily correlate acquired knowledge with the other subjects.
It strengthens the skill, complexity of practical subject and makes mastery over the practical subject.
It develops social relationship like – human and social qualities in students.
For teachers; it helps to complete the curriculum within short period of time and provides time for revision.
Knowledge is useful and is maintained so that it can be developed and used in day to day life.

Examples of dependent phenomena include the

Relation between parents and their offsprings.
The correlation between theprice and availabilty of product in the market.
The theoretical aspect of anything explains the practical concept.
Even a crime is also related to wrong addictions.
The negative correlation between age and normal vision.
The positive correlation between the incidence of lung cancer and cigarette smoking.

Correlation of Mathematics with other disciplines
Mathematics is “Science of all Sciences” and “Art of all Arts”. After understanding the basic concept of mathematics, students need to correlate the importance and concept of mathematics with other subjects, so as to understand other subjects easily and establishing relationship. Mathematical knowledge plays a crucial role in understanding the contents of other subjects.

MATHEMATICS WITH GENERAL SCIENCE: Science without mathematics is totally meaningless, because chemical reactions, scientific theories and detail of elements are only generated/ counted with the help of mathematics. Mathematics is used in most of applications like in work, energy, electricity, motion, gravitation, magnetism etc.

MATHEMATICS AND PHYSICS: child should have rich knowledge of mathematics to understand physics. Generally final shape to the rules of physics is given by mathematics; it presents these rules in practically workable form. Mathematical calculations occur in every step of physical science. Charle’s law of expansion of gases is based upon mathematical calculations, numerical problems on liquid, pressure,frictional force, laws of motion, gravitation, momentum etc.

MATHEMATICS AND CHEMISTRY: Molecular weights of organic compounds are calculated with mathematics. To measure the constituents of mixtures and Chemical compounds. To calculate Empirical or molecular formula. In balancing the chemical equations. In electronic configuration of atom of the element. Charle’s law of expansion of gases is based upon mathematical calculations.

MATHEMATICS AND BIOLOGY: Mathematics has very high correlation with biology. The Normal Weight, Caloric value, Rate of Respiration, Nutritive Value of Food, Transpiration, is calculated by Maths. The Growth in Weight of infants’ upto Nine months. To count the number of bones in human being and other different species. To measure blood pressure. To count the number of WBC & RBC in different blood groups. To count Sex chromosomes.

MATHEMATICS AND SOCIAL SCIENCES: After completion of the unit child can read, interpret, and draw the graphs. For example, to compare the Population- students can draw bar graphs, Population Density of various countries, Per Capita Income etc.

MATHEMATICS AND GEOGRAPHY: Geographical figures are explained in the terms of numbers only like seasonal conditions, temperature, humidity, degree, measurement of rain etc. the geographical conditions also defines the economy of a rich/poor country. Many countries like India have agricultural based economy due to its climate, rainfall, rivers and weather prediction.etc.

Certainly Mathematics is used for constituting the map, Formation of Nights & Days, Solar & Lunar Eclipse, Longitude Latitude, Maximum and Minimum Temperature, Barometric Pressure, Height above Sea Level, Surveying, Calculation of International, Local and Standard Time, Instruments etc. And here are also many other calculations. Punjab, Haryana and U.P are very fertile states in India, so contribute to grain stores, industries are established there but in these states there are no mines.

MATHEMATICS AND HISTORY: in history Mathematics helps in Calculation of Dates like duration of Britishers ruled in India? When Gandhi ji was born? Celebrate National Days and festivals, Cost in building of Taj-Mahal. Tenure of President in India. This gives us new information of the historical world. When the First and second world wars were fought? On account of economic considerations industrial revolution in Europe.

MATHEMATICS AND ECONOMICS: Statistical Methods are used to calculate and to know the Volume of Trade, Trend of Import and Exports, Economic Forecasts, Trade Cycles, It helps in calculating various indexes like crop production inflation, etc.

All economists, citizens and the businessmans can get the market trends & economic conditions. Through currencies market, the Current updates of currency and through stock and commodity market the current updates of the stock and commodity of different countries.
Only because of economic reasons certain empires faced liquidation. Similarly, economic events have been influenced or affectd by historical circumstances. In the current scenario the economic condition of India during UPA and NDA.

MATHEMATICS AND FINE ARTS: decides size, Ratio and Proportion while constructing the Similarity, Scale appreciation, Balance and Symmetry, Postulates, Drawing images on cloth and paper, Rhythm in Music etc.

MATHEMATICS AND LANGUAGE:

Math and Reading:- Students read about the discoveries or work of great mathematicians, and they can make poem on numbers.
Math and Writing (numbers are converted into writing):- A student makes the pie chart and interprets in his own words.

e.g. Counting of alphabet, vowel, Read About The Life History of Mathematicians. Student can draw make a bar graph of time spent in school and home the whole week and can interpret. (Interpretation of Non-Verbal Data)

MATHEMATICS WITH AGRICULTURE: Agriculture has close relationship with Maths. Agriculture has correlation with maths like area of crops, which season is suitable for which crop. How much quantity of water may be used in irrigation is also calculated in concern of agriculture by the use of mathematics. Investment, expenditure and saving in sowing specific crop, Division of land, Cost of labour, seed, fertilisers, expenditure in transportation of vegetables to the market, has the use of mathematics. As due to scientific inventions, there is lot of growth of agriculture & economy takes place.

MATHEMATICS WITH COMMERCE/ACCOUNTS: With the rich knowledge of commerce it is possible to study the economy of the country. Only by the knowledge of mathematics, Debit, Credit process & expenditure in accounts of industry, banks firm, etc are determined. The commerce teacher of should try to teachor make understand in such a way that students may relate and explain all specific terms mathematically.

MATHEMATICS AND ICT: The ICT is strongly correlated with mathematics. Computer programmes, applications, software and different languages without mathematics are impossible to operate and follow. Students are taught computers only because of knowledge of mathematics. Computer Provides important software for calculation e.g. SPSS software used in the long statistical calculations for research work. Many mathematical packages are used included Logo, dynamic geometry software, graph plotting etc., which are used in the teaching programmes.

MATHEMATICS WITH ENGINEERING: without mathematics Engineering is like sea without water. Mathematics has very strong correlation with each and every branch of engineering. Mathematics is used in every branch of engineering like Electronics, Electrical, Mechanical, Architect, Civil, Chemical, Computer etc. To get admission in any engineering stream, student must read Mathematics as a subject upto class 12.

MATHEMATICS WITH PSYCHOLOGY: Mathematics has correlation with Psychology for measuring I.Q, S.D, coefficient of correlation, Significance of difference, Measure of central tendency (Mean, Median, and Mode). “Likert Scale” used in psychology to make questionnaire. Mathematics is used in different modes of psychology like industry, army, social etc.

MATHEMATICS WITH ASTRONOMY: Counting of Stars and Planets, No. of moon/satellite of all planets. No. of stars in galaxy. Time taken in revolving at its own orbit. Formation of seasons, Life of star, galaxy etc. And Distance between two planets.

MATHEMATICS WITH PHYSICAL EDUCATION: Mathematics is used to measure structure of the body, blood pressure, the height, weight, rules of the games etc. Temperature of the normal human body, Size of playground, norm and standard of game like boot-ball, hockey, cricket, volleyball, tennis, wrestling, boxing etc.

One more important espect or field

MATHEMATICS WITH INDUSTRY: Mathematics is used in industrial work for example:- weaving, knitting, making furniture, leather work, making paints and fertilisers etc. Mathematical calculations are required to calculate all work and the cost.

References
http://www.businessdictionary.com/definition/correlation.html#ixzz2RmJKiVEA
http://en.wikipedia.org/wiki/Correlation_and_dependence
http://www.ditutor.com/regression/types_correlation.html
PPT by Ms. Namrata Katare Saxena, Asst Prof. PCER, MES
Dr. B. Pandya; Teaching of Mathematics (2007), Radha Prakhashan Mandir, Agra
S.K. Mangal; Teaching of Mathematics, Tondon Publications, Ludhiana