### 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*H*. The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality (=, â‰¤ or â‰¥)_{0} - Always write the
**alternative hypothesis**, typically denoted with*H*or_{a}*H*, using not equal, less than or greater than symbols, i.e., (â‰ , <, or > )._{1} - 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 *Î±* and *Î²*, for a Type I and a Type II error respectively. The power of the test, 1 â€“ *Î²*, 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 testâ€™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 success 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*and_{0}*H*. Remember, they are contradictory._{a} - 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*or cannot accept_{0}*H*), and write a clear conclusion using English sentences._{0}