Large sample tests, CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

In this section we describe and demonstrate the procedure for conducting a test of hypotheses about the mean of a population in the case that the sample size nn is at least 3030. The Central Limit Theorem states that  is approximately normally distributed, and has mean  and standard deviation , where μμ and σσ are the mean and the standard deviation of the population. This implies that the statistic

has the standard normal distribution, which means that probabilities related to it are given in Figure 7.1.5 and the last line in Figure 7.1.6.

If we know σσ then the statistic in the display is our test statistic. If, as is typically the case, we do not know σσ, then we replace it by the sample standard deviation ss. Since the sample is large the resulting test statistic still has a distribution that is approximately standard normal.

Standardized Test Statistics for Large Sample Hypothesis Tests Concerning a Single Population Mean

•  If σσ is known: 

• If σσ is unknown: 

The test statistic has the standard normal distribution.

The distribution of the standardized test statistic and the corresponding rejection region for each form of the alternative hypothesis (left-tailed, right-tailed, or two-tailed), is shown in Figure 8.2.18.2.1.

Figure 8.2.18.2.1: Distribution of the Standardized Test Statistic and the Rejection Region