Model and hypotheses
In both examples in the first page of this section, there was knowledge of the population standard deviation σ (at least when H0 was true). This greatly simplifies the problem of finding a p-value for the test.
In both examples, the hypotheses were of the form,
H0 : μ = μ0
HA : μ ≠ μ0
Summary Statistic
The first step in finding a p-value for the test is to identify a summary statistic
that throws light on whether H0 or HA is true.
When testing the population mean, µ,
the obvious summary statistic is the sample mean, ,
and the hypothesis tests that will be described here are based on this.
We saw earlier that sample mean has a distribution with mean and standard deviation
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= μ |
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= | ![]() |
Furthermore, the Central Limit Theorem states that the distribution of the sample mean is approximately normal, provided the sample size is not small. (The result holds even for small samples if the population distribution is also normal.)
P-value
The p-value for the test is the probability of getting a sample mean as 'extreme' as the one that was recorded when H0 is true. It can be found directly from the distribution of the sample mean.
Note that we can assume knowledge of both µ and σ in this calculation — the values of both are fixed by H0
Since we know the distribution of the sample mean (when H0 is true), the p-value can be evaluated as the tail area of this distribution.