7.1 Null and Alternative Hypotheses
In a hypothesis test, sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:
- Evaluate the null hypothesis, typically denoted with \(H_0\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, ≤ or ≥)
- Always write the alternative hypothesis, typically denoted with \(H_a\) or \(H_1\), using not equal, less than or greater than symbols, i.e., (≠, <, or > ).
- If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
- Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.
7.2 Outcomes and the Type I and Type II Errors
A Type I error occurs when a true null hypothesis is rejected. A Type II error occurs when a false null hypothesis is not rejected.
The probabilities of these errors are denoted by the Greek letters \(\alpha\) and \(\beta\), for a Type I and a Type II error respectively. The power of the test, \(1 – \beta\), quantifies the likelihood that a test will yield the correct result of a false null hypothesis being rejected. A high power is desirable.
7.3 Distribution Needed for Hypothesis Testing
In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.
When testing for a single population mean:
- A Student's \(t\)-test should be used if the data come from a random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
- The normal test will work if the data come from a random sample and the population is approximately normally distributed, or the sample size is large.
When testing a single population proportion use a normal test for a single population proportion if the data comes from a random sample with \(n\) > 100.
7.4 Full Hypothesis Test Examples
The hypothesis test itself has an established process. This can be summarized as follows:
- Determine \(H_0\) and \(H_a\). Remember, they are contradictory.
- Determine the random variable.
- Determine the distribution for the test.
- Draw a graph and calculate the test statistic.
- Compare the calculated test statistic with the \(z\) critical value determined by the level of significance required by the test and make a decision (cannot reject \(H_0\) or reject \(H_0\)), and write a clear conclusion using full sentences.