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Indeedthe Pearson r product-moment correlation coefficient is very usefulin describing the extent of any statistical relationship especially,when two intervals are provided (Wu et al., 2016). The Pearson canshow significance or fail to but when none of it exist, finding thegradient of the linear trend line is hard. There is statisticalsignificance when r falls in the rejection part and in such a casethe null hypothesis may not be accepted (Huck et al., 2013). If the ris close to zero, it means that the relationship is very minimal, butit is adamant when it is farther away. The use of scatterplotdiagrams can show the type of relationship that exists between thetwo variables, which can either be positive, negative, or absent.However, it is normal to expect a weak correlation if a large samplesize was used (Giroldini et al., 2016).

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Typically,in testing the hypothesis, the null one is assumed to be true (Ruxtonet al., 2015). It is valid to state that the significant linearcorrelation do not exist and this is applicable where there is afailure to reject the null hypothesis provided the critical valueexceeds the test static (Romano et al., 2011). The dependentvariables can be affected by the independent ones, but such an impactshould have statistical significance (von Toussaint et al., 2012).Nevertheless, if such importance were to be present, the linearcorrelation would be positive, and the null hypothesis would berejected. In this case, the critical value becomes small compared tothe test statistic, which turns out to be positive. Although theinitial hypothesis is only concerned with the positive correlation,it is possible for the negative one to exist because each of them hasequal chances and are valid. The relationship between the GRE and GPAcan, therefore, take any form since it is not fixed.

References

Giroldini,W., Pederzoli, L., Bilucaglia, M., Melloni, S., &amp Tressoldi, P.(2016). A new method to detect event-related potentials based onPearson`s correlation. EURASIPJournal on Bioinformatics &amp Systems Biology,2016(1),1-13. doi: 10.1186/s13637-016-0043-z.

Huck,S. W., Bixiang, R., &amp Hongwei, Y. (2013). A New Way to Teach (orCompute) Pearson`s r Without Reliance on Cross-Products. TeachingStatistics,29(1),13-16. doi:10.1111/j.1467-9639.2007.00240.x.

Romano,J.P., Shaikh, A.M &amp Wolf, M. (2011). Hypothesis Testing inEconometrics. Annual Review of Economics, Vol. 2, pp. 75-104retrieved from http://www.jstor.org/stable/42940324on October 17, 2016.

Ruxton,G. D., Wilkinson, D. M., &amp Neuhäuser, M. (2015). Advice ontesting the null hypothesis that a sample is drawn from a normaldistribution. AnimalBehaviour,107249-252.doi:10.1016/j.anbehav.2015.07.006.

vonToussaint, U., Frey, M., &amp Gori, S. (2012). Fitting of Functionswith Uncertainties in Dependent and Independent Variables. AIPConference Proceedings,1193(1),302-310. doi:10.1063/1.3275627.

Wu,K., &amp Li, W. (2016). On a dispersion model with Pearson residualresponses. ComputationalStatistics &amp Data Analysis,10317-27.doi:10.1016/j.csda.2016.04.015.