6.5.5—Normal distribution

For X~N(μ,σ²), standardise observations or sample means with z=(x−μ)/σ and use the standard normal tail. A two-sided alternative uses both tails.

Write the null parameter and variance clearly, draw the rejection region and convert back to the original variable when reporting a critical boundary.

Testing μ=100 with σ=15 and n=25 uses standard error 3, so x̄=106 gives z=2.

The test statistic’s denominator is the standard error for a mean, not the individual standard deviation σ.