In inferential statistical analysis the researcher studies a sample about a variable or correlation between variables and then infers from the sample and tests statistically whether it is applicable to the general population. That is, the researcher infer from the sample whether the relationship is due to chance or the relationship is significant at a particular level of probability level of error acceptable given the type of phenomenon under study.  

 In this essay I will discuss how statistical significance is applied and the use of theoretical probability distribution is used for larger and smaller samples to test the null hypothesis and conclude after calculating the standard error or the probability of that standard error or more or less and compare with the level of probability of significance and then conclude whether there exists a relationship in the population from the sample studied.

The Importance of sample size in testing significance of a correlation coefficient at 0.05 accepted random probability Level

Say that a researcher is testing a correlation between two variables correlation co-efficient and set a sample size of n=5 to n=30 and graph and wants to check the probability of r< -0.5 and r>0.5 due to randomness. In this exercise he will discover when the sample size is 5 the percentage of cases at a particular r is not closer to the theoretical distribution of normal distribution. However, when the sample size increases, he will see the graph of r and the percentage of the r occurring will move towards the theoretical normal distribution. As well, the probability of r greater than +0.5 or r less than 0.5, tends to decrease to approximately 0.25 when the sample size is equal to 30. That means when sample size increases the probability of a particular positive r greater than or negative r less than tends to reduce due to randomness. There fore if the sample size is quite large then the sample tends to move towards the theoretical normal distribution and it can be used to predict whether a particular value at a particular acceptable level of probability error is significant or not depending on the probability of that figure is less than or greater than the 0.05 level of significance. If the probability is less than 0.05 then the r will be significant if the probability is greater then it is not significant at 0.05 levels.

The Process of Significance testing regarding the testing of the correlation co-efficient

To test the significance statistically one first has to set the significance level acceptable depending on the accuracy level accepted for that particular research. This is normally at 0.05 level of probability. However, in some instances if more accuracy is needed then it can be at 0.01 or less than 0.05 levels.

The second step is to set the sample size which is large enough to represent the population to have a feasible prediction as discussed above. The sample size is dependent on cost factors, time as well as the impact of sample size on the assumption of normal curve of probability distribution and the reduction of standard error.

The third step is to determine to test directional or non directional significance testing. In relation to correlation significance is to test if the correlation is greater then a positive figure that is directional or ignoring the sign whether it is more than the positive correlation or less than the negative correlation. This is non-directional.

The fourth step is to cal calculate test t value for a particular r. The t-value for a small sample is equal to the following formula

The t- value = r/square root of [(1-r2)/ (N-2) where N-2 is the degree of freedom

The fifth step is to use the t-distribution table using the degree of freedom significance level to find the critical t- value or the p-value which is the probability of the r greater than or different than r if it is non-directional.

The last stage is the inference stage. In this last stage compare the p-value with the significance level. If the p-value is less than the significance level and the sample size is sufficient enough then the r is significant. If the p value is greater than the significance level then the r or the correlation co-efficient is not significant.

For example say a researcher has set n= 50 and the level of significance at 0.05 level and he wants to test whether the correlation co-efficient from this sample of +0.87 in a non directional null hypothesis whether it is significant. If the sample size is large enough and it is normally distributed then the t- value applying the above formula will be 12.225. At this level the non directional probability is less than 0.000001. This is less than 0.05. There fore the correlation coefficient is significant. That is the r is not due to chance at this level of acceptable random error and the sample correlation co-efficient of 0.87 is a reasonable estimate of the co=relation coefficient of the population or in general.

Conclusion

As discussed above the statistical significance is an important one because it test whether the co-relation coefficient of the sample is due to chance and there fore it cannot be inferred with confidence for the population or the r calculated by the sample is significant depending on the p-value and the sample size and it satisfies the theoretical normal distribution assumptions.

 If the significance level is not appropriate or the sample size is very small then one must be careful to infer from the sample to the population because it will be misleading and to conclude erroneously and it may exist because of fluke or chance. The Significance mechanics is not important but the logic of analysis and the relevance of the theoretical distribution depending on the variable and the selection of sample and the determination of the sample size and the application of the relevant theoretical probability distribution.