Testing Hypotheses

Definitions

• A statistical hypothesis is an assertion of conjecture concerning one or more populations
• Rejection of the null hypothesis when it is true is a Type I error
• The probability of committing a Type I error is called the level of significance denoted by $\alpha$
• Acceptance of a null hypothesis when it is false is a Type II error
• The power of a test is the probability of rejecting $H_{0}$ given that a specific alternative is true
• A $P$ is the lowest level (of significance) at which the observed value of the test is significant. It is equal to the significance level of the test for which we would only just reject the null hypothesis. The $P$ value is compared with the actual significance level of our test and, if it is smaller, the result is significant. That is, if the null hypothesis were to be rejected at the 5% signficance level, this would be reported as $\mathbf{P < 0.05}$

  Small $P$ values suggest that the null hypothesis is unlikely to be true. The smaller it is, the more convincing is the rejection of the null hypothesis. It indicates the strength of evidence for say, rejecting the null hypothesis $H_{0}$, rather than simply concluding Reject $H_{0}$ or Do not reject $H_{0}$

General Procedeure for Hypothesis Testing

We assume that the hypothesis is $H_{0}: \theta = \theta_{0}$

1. State the null hypothesis $H_{0}$ that $\theta = \theta_{0}$
2. Choose the appropriate alternative hyppthesis, $H_{1}$ from one of the alternatives $\theta < \theta_{0}$, $\theta > \theta_{0}$, $\theta \not= \theta_{0}$
3. Choose a significance level of size $\alpha$
4. Select the appropriate test statistic and establish the critical region. (If the decision is to be based on a $P$ value then it is not necessary to state the critical region)
5. Compute the value of the test statistic from the sample data
6. Decision: Reject $H_{0}$ is the test statistic has a value in the critical region (or if the computed $P$ value is less than or equal to the desired significance level $\alpha$); otherwise, do not reject $H_{0}$