Large Sample Test for Mean
A random sample of size n (n > 30) has a sample mean . To test the hypothesis that the population mean μ has a specified value μ_{0} let us formulate the Null Hypothesis as H_{0}: μ = μ_{0} .
The Alternative Hypothesis is (i) H1: μ ≠ μ_{0} or (ii) H1: μ > μ_{0} or (iii) H1: μ < μ_{0} . Since n is large, the sampling distribution of is approximately normal.
If H0 is true, the test statistic z = has approximately a standard normal distribution. The critical region for z depending on the nature of H1 and level of significance α is given below:
Level of significance ( α )
10%
5%
1%
Critical region for μ ≠ μ_{0}
| z | > 1.64
| z | > 1.96
| z | > 2.58
Critical region for μ > μ_{0}
z < -1.28
z < -1.64
z < -2.33
Critical region for μ < μ_{0}
z > 1.28
z > 1.64
z > 2.33