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Assume a sample of size n is drawn from a population with mean m and S.D. s, so that the sample mean  has mean m and S.D. s/. We test

H₀: m = m₀ vs. H_a: m¹m₀ OR m>m₀ OR m<m₀

CASE 1: Normal population with s known.

Test Statistic is z=

H_a: m >m₀Þ Reject if z³ z_a
H_a: m <m₀Þ Reject if z£-z_a
H_a: m¹m₀Þ Reject if z³z_a/2 OR z£-z_a/2, i.e. |z|³z_a/2

CASE 2: Arbitrary population with large sample size n.
Test Statistic is z= if s is known and  otherwise.
        Rejection regions are same as in Case 1.
 

CASE 3: Normal population with s unknown and small sample.
Test Statistic is t =

H_a: m >m₀Þ Reject if t ³t_a,n-1
H_a: m <m₀Þ Reject if t £-t_a,n-1
H_a: m¹m₀Þ Reject if t³ t_a/2,n-1 OR t£-t_a/2,n-1, i.e. |t|³t_a/2,n-1

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