A **statistical hypothesis** is an assumption about a **population parameter**. For example, we may assume that the mean height of a male in the U.S. is 70 inches. The assumption about the height is the* statistical hypothesis* and the true mean height of a male in the U.S. is the* population parameter*.

A **hypothesis test** is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

**The Two Types of Statistical Hypotheses**

To test whether a statistical hypothesis about a population parameter is true, we obtain a random sample from the population and perform a hypothesis test on the sample data.

There are two types of statistical hypotheses:

The **null hypothesis**, denoted as H_{0}, is the hypothesis that the sample data occurs purely from chance.

The **alternative hypothesis**, denoted as H_{1} or H_{a}, is the hypothesis that the sample data is influenced by some non-random cause.

**Hypothesis Tests**

A **hypothesis test** consists of five steps:

**1. State the hypotheses. **

State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false.

**2. Determine a significance level to use for the hypothesis.**

Decide on a significance level. Common choices are .01, .05, and .1.

**3. Find the test statistic.**

Find the test statistic and the corresponding p-value. Often we are analyzing a population mean or proportion and the general formula to find the test statistic is: (sample statistic – population parameter) / (standard deviation of statistic)

**4. Reject or fail to reject the null hypothesis.**

Using the test statistic or the p-value, determine if you can reject or fail to reject the null hypothesis based on the significance level.

The *p-value* tells us the strength of evidence in support of a null hypothesis. If the p-value is less than the significance level, we reject the null hypothesis.

**5. Interpret the results. **

Interpret the results of the hypothesis test in the context of the question being asked.

**The Two Types of Decision Errors**

There are two types of decision errors that one can make when doing a hypothesis test:

**Type I error:** You reject the null hypothesis when it is actually true. The probability of committing a Type I error is equal to the significance level, often called *alpha*, and denoted as α.

**Type II error:** You fail to reject the null hypothesis when it is actually false. The probability of committing a Type II error is called the Power of the test or *Beta*, denoted as β.

**One-Tailed and Two-Tailed Tests**

A statistical hypothesis can be one-tailed or two-tailed.

A **one-tailed hypothesis** involves making a “greater than” or “less than ” statement. For example, suppose we assume the mean height of a male in the U.S. is greater than or equal to 70 inches. The null hypothesis would be H0: µ ≥ 70 inches and the alternative hypothesis would be Ha: µ < 70 inches.

A **two-tailed hypothesis** involves making an “equal to” or “not equal to” statement. For example, suppose we assume the mean height of a male in the U.S. is equal to 70 inches. The null hypothesis would be H0: µ = 70 inches and the alternative hypothesis would be Ha: µ ≠ 70 inches.

*Note:* The “equal” sign is always included in the null hypothesis, whether it is =, ≥, or ≤.