We wish to test a hypothesis about the mean of a population, µ. For example, suppose our hypotheses are:
.
We have a random sample from the population, . A good estimate µ is
, so this sample mean will form the basis of our test. We choose a significance level, α. In this example, we'll use α = 0.08; the chance that we make a type I error is then 8%. Let's assume we have N = 100
The form of the test depends on what other assumptions we make.
Population is normal; the population standard deviation, s is known.
In this case, has a normal distribution with a mean of µ=10 and a standard deviation of
. Let's suppose that σ = 2.
The test statistic has a normal distribution with a mean of zero and a standard deviation of one.
We'll reject H0 in favor of HA if
Since α = 0.08, zα = 1.75. So we reject H0 if:
.
We can write the conditions for rejecting H0 as:
.
We fail to reject H0 if:
.