Level of Significance: α

The main purpose of hypothesis testing is not to question the computed value of the sample statistic, but to make judgment about the difference between the sample statistic and a hypothesized population parameter. The next step after stating the Null and Alternative Hypotheses, is to decide what criterion to be used for deciding whether to accept or reject the null hypothesis.

When we choose 5% level of significance in a test procedure, there are about 5 cases in 100 that we would reject the hypothesis when it should be accepted, that is, we are about 95% confident that we have made the right decision. Similarly, if we choose 1% level of significance in testing a hypothesis, then there is only 1 case in 100 that we would reject the hypothesis when it should be accepted.

Suppose, that under a given hypothesis the sampling distribution of a statistic θ is approximately a normal distribution with mean

E (θ) and standard deviation (Standard Error) σ_θ

Figure 

Then z

=

is called the standardized normal variable or z-score, and its distribution is the standardized normal distribution with mean 0 and standard deviation 1, the graph of which is shown above.

From the above figure, we see that if the test statistic z of a sample statistic  θ lies between -1.96 and 1.96, then we are 95% confident that the hypothesis is true [since the area under the normal curve between z = -1.96 and z  = 1.96 is 0.95 which is 95% of the total area].

But if for a simple random sample we find that the test statistic (or z-score) z lies outside the range -1.96 to 1.96, i.e. if z  > 1.96, we would say that such an event could happen with probability of only 0.05 (total shaded area in the above figure if the given hypothesis were true). In this case, we say that z-score differed significantly from the value expected under the hypothesis and hence, the hypothesis is to be rejected at 5% (or 0.05) level of significance. Here the total shaded area 0.05 in the above figure represents the probability of being wrong in rejecting the hypothesis. Thus if z  > 1.96, we say that the hypothesis is rejected at a 5% level of significance.

The set of z scores outside the range -1.96 and 1.96, constitutes the critical region or region of rejection of the hypothesis or the region of significance. Thus critical region is the area under the sampling distribution in which the test statistic value has to fall for the null hypothesis to be rejected. On the other hand, the set of z scores inside the range -1.96 to 1.96 is called theregion of acceptance of the hypothesis. The values -1.96 and 1.96 are called critical values at 5% level of significance.

From the above discussion we can formulate the following rule of decision:

Decision Rule (Two-Sided Tests)