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- Effect Size
The effect size that is associated to the data is dependent on the type of test that is involved in analyzing the sample. In this particular case, the test ran were Bootstrapped correlation and randomized correlation. The effect size is considered as how big difference is made by intervention. It the effect size, both statistical and clinical significances are not great however small correlations around 0.20 require larger sample sizes, whereas medium correlations like 0.40 require medium sized sample size. For correlations that are larger and around about 0.60, smaller sample sizes are required. As for the given correlation, the value is about 0.60 therefore a sampler sample size would work effectively, therefore it is suggested that n=25.
- Desired Power
Usually powers have the ability to detect the difference between the mean scores and the magnitude of correlation however, if there is not enough power in the study, this certainly is not affected by the effect size, as there are many studies which are under powered. Once the effect size is determined, it is only then effective to analyze what the power level of the data is. It is possible to state a certain power level that is needed to be determined with the sample but that power level might not work effectively with the effect size of the sample. Therefore in order to get the desired 95% of power it is recommended to have 250 samples. However as the effect size is large d=0.60, therefore it is recommended to have smaller sample size i.e. 25.
The figure 1 shows the results of drawing 1000 bootstrapped samples, each of n = 50, with replacement from the original data set. As you can see, the sampling distribution of r is positively skewed, as we would expect. The 95% CI are given as .548 and .792. Those are fairly wide intervals, but n = 50, which is not very large for setting confidence limits. In addition, confidence limits have an unpleasant habit of generally being larger than we would like. Notice that the limits do not include 0.0, which confirms that the correlation is significant, using a test that does not rely on bivariate normality of the data. By the nature of the variables used in this study, it is reasonable to expect that an assumption of bivariate normality is not terribly unreasonable.
In the figure 2 you can see that the correlation obtained by Katz et al. (1990) on the original data was .686. You can also see that the sampling distribution of r under randomization is symmetrical around 0.0, and that 0 of the 10,000 randomizations exceeded +.686. This gives us a probability under the null of .000, which will certainly allow us to reject the null hypothesis. This is a two-tailed test, and, because the distribution is symmetric for p = 0, you will not go far wrong if you cut the probability in half for a one-tailed test.
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