Statistical Analysis for Education and Psychology Researchers

As with previous tests of significance the sampling distribution of the test statistic is
used to determine whether the observed statistic is likely to have occurred by chance.
Unfortunately, the sampling distribution of r is not normal in form when sample sizes are
small (n<30) and when the population correlation deviates from zero. To overcome this
problem r is transformed, and the probability of this estimator is based on the sampling
distribution of the t-statistic. The significance of an obtained Pearson correlation is
therefore evaluated using the t-distribution with n−2 degrees of freedom and is given by
the following equation:

Significance

of r—8.12

The null hypothesis tested is that the two variables are independent, that is there is no
linear relationship between them, H 0 :ρ=0, and the rejection region (for the null
hypothesis) is /t| (the absolute value of t)>t 1 −α/2. The alternative hypothesis is, H 1 :ρ≠0, and
the rejection region (one sided-tests) is t<−t 1 −α or t>t 1 −α.
To answer the question, Is there a significant correlation, at the 5 per cent level,
between MATHS and RAVEN scores? t would be calculated as follows:

Interpretation

The critical t-value from Table 3, Appendix A4 is 2.306. The test statistic does not
exceed this critical value, (2.070<2.306), and the null hypothesis is therefore not rejected.
We would conclude that the correlation is not significant at the 5 per cent level.

Confidence Intervals for Correlation r

A 95 per cent confidence interval for the population correlation allows the accuracy,
hence trustworthiness of any statistical inferences to be estimated. The confidence
interval is based on a transformation of the statistic r to a statistic called Fisher’s z. This
is not the same as the Z-deviate from a normal distribution (sometimes called a Z-score).
To interpret the confidence interval the Fisher’s z-score has to be transformed back to the
correlation metric. Fisher’s z is evaluated as,

(SE is standard error)
The 95 per cent CI for the bivariate correlation between SMATHS and MATHS
(r=0.896) is

Statistical analysis for education and psychology researchers 288