Hypothesis Testing

In hypothesis testing, the \(\mathbf{p}\)-value is generally defined as:

The probability, assuming the null hypothesis is true, of obtaining a result as extreme as or more extreme than the observed sample result.

In context, you should always mention:

1. Assume \(H_0\) is true
2. Probability of getting the observed test statistic or something more extreme
3. Use the direction of the alternative hypothesis

For example:

Right-tailed test

If

\[H_0 : \mu = 50, \quad H_1 : \mu > 50\]

and the sample mean is larger than 50, then the p-value is:

The probability, assuming the true mean is 50, of obtaining a sample mean at least as large as the one observed.

So "more extreme" means larger.

Left-tailed test

If

\[H_0 : p = 0.3, \quad H_1 : p < 0.3\]

then the p-value is:

The probability, assuming the true proportion is 0.3, of obtaining a sample proportion at most as small as the one observed.

So "more extreme" means smaller.

Two-tailed test

If

\[H_0 : \mu = 20, \quad H_1 : \mu \neq 20\]

then the p-value is:

The probability, assuming the true mean is 20, of obtaining a sample mean as far from 20 as, or further than, the observed sample mean.

So "more extreme" means further away from the null value in either direction.

A good general template is:

The p-value is the probability, assuming \(H_0\) is true, of obtaining a test statistic at least as extreme as the observed value in the direction of \(H_1\).

For exam writing, avoid saying "the p-value is the probability that \(H_0\) is true." That is incorrect.