## 9.1 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 H0. 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., (\(neq\), <, 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.

## 9.2 Outcomes and the Type I and Type II Errors

In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or misunderstood summary statistics can yield errors that affect the results. 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 true alternative hypothesis being accepted. A high power is desirable.

## 9.3 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 simple, 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 simple, 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 simple, random sample, fill the requirements for a binomial distribution, and the mean number of successes and the mean number of failures satisfy the conditions: \(np > 5\) and \(nq > 5\) where \(n\) is the sample size, \(p\) is the probability of a success, and \(q\) is the probability of a failure.

## 9.4 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 cannot accept \(H_0\)), and write a clear conclusion using English sentences.