Correlation and the Coefficient of Determination - Essay Example

Pearson’s correlation indicates the strength of the relationship between two sets of variables. A direct relationship indicates that as A increases B increases with it and vice versa; while a negative relationship indicates that as A increases B decreases and vice versa. If r = 0, there is no relationship.

The coefficient of determination is found by squaring the coefficient of correlation. According to Webster.edu (n.d.), it is a measure of the proportion of the variance in B that can be explained by knowing A and vice versa and provides one of the best means of evaluating the strength of the relationship between two variables.

In order to determine whether there is a correlation between the two sets of test scores, we would first need to state the null hypothesis that is to be tested. According to Mason and Lind (1996), this hypothesis is called the null hypothesis and is designated H0. There is also an alternative hypothesis that is designated H1; this indicates what would be the conclusion if the null hypothesis is rejected. In this case, the null hypothesis would indicate that there is no correlation between both test scores, and in the alternative, it would indicate that a correlation exists. The null and alternative hypothesis is stated as follows.

H0: µ = 0
H1: µ ≠ 0
We now need to determine alpha which is the level of significance. Mason and Lind indicate that the level of significance is the probability of rejecting the null hypothesis when it is actually true and also states that this would represent a type 1 error. When alpha (α) is equal to 0.05 it indicates that there is a 95% confidence level. In other words, alpha is the critical value and is the probability that in accordance with the null hypothesis a statistical test will generate a type 1 error.

The p-value is a test statistic that is compared with the critical value in order to determine whether to accept or reject the null hypothesis. The null hypothesis is rejected the p-value is less than 0.05 (p < 0.05).

In the table below Multiple R is the correlation coefficient, R square (R2) is the coefficient of determination and the adjusted R2 square is R2 adjusted for the degrees of freedom. The number of test scores which is the number of observations was 62.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.25155255
R Square 0.06327869
Adjusted R Square 0.04766666
Standard Error 19.988518
Observations 62

ANOVA
df SS MS F Significance F
Regression 1 1619.419953 1619.419953 4.053202449 0.048580252
Residual 60 23972.45102 399.5408503
Total 61 25591.87097

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 54.0702033 6.564506775 8.23675033 1.93538E-11 40.93923486 67.201172 40.939235 67.20117183
A 0.22947081 0.11397991 2.013256677 0.048580252 0.001477051 0.4574646 0.0014771 0.457464578

In the table, the results indicate that R which is the correlation coefficient is equal to 0.25 and the coefficient of determination is equal to 0.063. This suggests that there is some correlation between test scores A and B.

In order to determine whether the correlation is significant the computed p-value is compared with the nominal alpha value. The nominal alpha which is set at 0.05 is less than the actual p statistic then the difference is significant. When alpha = 0.05 there is a 95% level of confidence that the scores will lie between 41 and 67%.

The test result in the table shows that the p-value is 1.935. This value is greater than alpha which is 0.05. Since the p statistic is greater than 0.05 we accept H0 and indicate that the level of correlation is not statistically significant.