9 Hypothesis Testing

There are slides and a YouTube Video associated with this chapter:

• Hypothesis Testing - Slides
• Hypothesis Testing - Video

A hypothesis is a statement about a population claiming that a parameters takes on a particular value or falls within a certain range of values. The steps of a hypothesis test are:

1. Formulating the null hypothesis \(H_0\) stating that the parameter takes a particular value:
   • One-sided test: \(H_0\): \(\mu \geq \mu_0\) or \(\mu \leq \mu_0\)
   • Two-sided test: \(H_0\): \(\mu = \mu_0\)
2. Setting the significance level \(\alpha\), e.g., 1%, 5%, or 10%.
3. Test statistic: Value based on the sample used to reject or fail to reject the null hypothesis.
4. Critical value and p-value:
   • Critical value represents the border point between rejecting and failing to reject \(H_0\).
   • p-Value: Probability of observing the parameter given the null hypothesis. Small p-values represent evidence against \(H_0\).

Note that equality is always part of \(H_0\), i.e., \(=\), \(\leq\), or \(\geq\).

There are two types of errors associated with hypothesis testing. A Type I error occurs if \(H_0\) is true but you reject \(H_0\). A Type II error occurs if \(H_a\) is true but you fail to reject \(H_0\). The significance level is the largest acceptable probability of committing a Type I error. This is denoted with \(\alpha\). There are two-sided and one-sided hypothesis tests: