### 9.1 Null and Alternative Hypotheses

In a **hypothesis test**, sample data are 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 do the following:

- 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 (=, ≤, or ≥)._{0} - Always write the
**alternative hypothesis**, typically denoted with*H*or_{a}*H*, using less than, greater than, or not equals symbols, i.e., (≠, >, 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 *α* and *β*, for a Type I and a Type II error respectively. The power of the test, 1 – *β*, 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, with a known standard deviation.

When testing a single population proportion use a normal test for a single population proportion if the data come 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* > *n* where *n* is the sample size, *p* is the probability of a success, and *q* is the probability of a failure.

### 9.4 Rare Events, the Sample, and the Decision and Conclusion

When the probability of an event occurring is low, and it happens, it is called a rare event. Rare events are important to consider in hypothesis testing because they can inform your willingness not to reject or to reject a null hypothesis. To test a null hypothesis, find the *p*-value for the sample data and graph the results. When deciding whether or not to reject the null the hypothesis, keep these two parameters in mind:

*α*>*p*-value, reject the null hypothesis.*α*≤*p*-value, do not reject the null hypothesis.

### 9.5 Additional Information and 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, calculate the test statistic, and use the test statistic to calculate the
*p*-value. (A*z*-score and a*t*-score are examples of test statistics.) - Compare the preconceived
*α*with the*p*-value, make a decision (reject or do not reject*H*), and write a clear conclusion using English sentences._{0}

Notice that in performing the hypothesis test, you use *α* and not *β*. *β* is needed to help determine the sample size of the data that are used in calculating the *p*-value. Remember that the quantity 1 – *β* is called the **Power of the Test**. A high power is desirable. If the power is too low, statisticians typically increase the sample size while keeping *α* the same. If the power is low, the null hypothesis might not be rejected when it should be.