Inferential statistics deal with inferences. Inferential statistics are concerned with determining how likely it is that results based on a sample or samples are the same results that would have been obtained for the entire population. Inferences about populations based on the behaviour of samples.
Concept of Standard Error
If we randomly select a number of samples from the same population and compute the mean for each it is likely that each mean will be somewhat different from each other mean, and that none of the means will be identical to the population mean. The variation among the means is referred to as sample error. If a difference is found between sample means, the question of interest is whether the difference is a result of sampling error or a reflection of a true difference.
Sample Size and Standard Error
Sample errors are interesting. They are normally distributed and most of the sample-means will be very close to the population mean; the number of means which are considerably different from the population mean will decrease as the size of the difference increases.
Standard deviation of the sample means (the standard deviation of sampling errors) is usually referred to as the standard error of the mean. The standard error of the mean tells us by how much we would expect our sample means to differ if we used other samples from the same population. According to normal curve percentages, we can say that approximately 68% of the sample means will fall between plus and minus one standard error of the mean, 95% will fall between plus and minus two standard errors, and 99+% will fall between plus and minus three standard errors.
If we know the standard deviation, then the standard error of the mean is equal to the standard deviation divided by the square root of the sample size. SE(mean) = SD/square root of (N-1). If a sample mean is 80, and the SE mean is 1.00, if we say that the population mean falls between 79 and 81, we have approximately 68% chance of being correct, if we say that the population mean falls between 78 and 82, we will have approximately a 95% chance of being correct, if we say that the population mean falls between 77 and 83, we will have approximately 99+% chance of being correct. In another word, the probability of the population mean being less than 77 and larger than 83 is less than 1%.
It is obvious now that a smaller standard error indicates less sampling error. The major factor affects standard error of the mean is sample size. The size of the sample increases the standard error of the mean decreases. Another factor affecting the standard error of the mean is the size of the population standard deviation. If the population standard deviation is large, members of the population are very spread out on the variable of interest, and the sample means will also be very spread out.
In order to determine whether or not the difference between those means probably represents a true population difference, we need an estimate of the standard error of the difference between two means.
Test Null Hypothesis
When we talk about the difference between two sample means being a true difference we mean that the difference was caused by the treatment and not by chance. The chance explanation for the difference is called the null hypothesis. The null hypothesis says in essence that there is no difference or relationship between parameters in the populations and that any difference or relationship found for the samples is the result of sampling error. The research hypothesis usually states that one method is expected to be more effective than another. Utilizing null hypothesis is more conclusive support for a positive research hypothesis. Suppose one hypothesizes that all research textbooks contain a chapter on sampling. If he or she examines and finds that a book does contain the chapter, it does not approve the hypothesis, because it is only one book. In other word, if he or she finds a book does not contain the chapter, it is enough to disapprove the hypothesis. The result of a study can reject the null hypothesis or not reject the null hypothesis. If it is rejected, the hypothesis PROBABLY false, if it is not rejected, the hypothesis PROBABLY true.
Test of Significance
In order to test a null hypothesis we need a test of significance and we need to select a probability level that indicates how much risk we are willing to take that the decision we make is wrong. At the end of an experimental research study, if there is a little difference between the group means, then researcher needs to decide whether the difference is significant or different enough to conclude that they represent a true difference. The test of significance is made at a pre-selected probability level and allows the researcher to state that he has rejected the null hypothesis. The level will be usually set at 0.05, or 0.01. That means the researcher will have 5% or 1% times to find the difference by chance. There are a number of different tests of significance that can be applied in research studies, t-test, analysis of variance and chi square etc.