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*.

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.

Whenever we perform a hypothesis test, we always write down a null and alternative hypothesis:

**Null Hypothesis (H**The sample data occurs purely from chance._{0}):**Alternative Hypothesis (H**The sample data is influenced by some non-random cause._{A}):

A hypothesis test can either contain a directional hypothesis or a non-directional hypothesis:

**Directional hypothesis:**The alternative hypothesis contains the less than (“<“) or greater than (“>”) sign. This indicates that we’re testing whether or not there is a positive or negative effect.**Non-directional hypothesis:**The alternative hypothesis contains the not equal (“≠”) sign. This indicates that we’re testing whether or not there is some effect, without specifying the direction of the effect.

Note that directional hypothesis tests are also called “one-tailed” tests and non-directional hypothesis tests are also called “two-tailed” tests.

Check out the following examples to gain a better understanding of directional vs. non-directional hypothesis tests.

**Example 1: Baseball Programs**

A baseball coach believes a certain 4-week program will increase the mean hitting percentage of his players, which is currently 0.285.

To test this, he measures the hitting percentage of each of his players before and after participating in the program.

He then performs a hypothesis test using the following hypotheses:

**H**μ = .285 (the program will have no effect on the mean hitting percentage)_{0}:**H**μ > .285 (the program will cause mean hitting percentage to increase)_{A}:

This is an example of a **directional hypothesis** because the alternative hypothesis contains the greater than “>” sign. The coach believes that the program will influence the mean hitting percentage of his players in a *positive* direction.

**Example 2: Plant Growth**

A biologist believes that a certain pesticide will cause plants to grow less during a one-month period than they normally do, which is currently 10 inches.

To test this, she applies the pesticide to each of the plants in her laboratory for one month.

She then performs a hypothesis test using the following hypotheses:

**H**μ = 10 inches (the pesticide will have no effect on the mean plant growth)_{0}:**H**μ < 10 inches (the pesticide will cause mean plant growth to decrease)_{A}:

This is also an example of a **directional hypothesis** because the alternative hypothesis contains the less than “<” sign. The biologist believes that the pesticide will influence the mean plant growth in a *negative* direction.

**Example 3: Studying Technique**

A professor believes that a certain studying technique will influence the mean score that her students receive on a certain exam, but she’s unsure if it will increase or decrease the mean score, which is currently 82.

To test this, she lets each student use the studying technique for one month leading up to the exam and then administers the same exam to each of the students.

She then performs a hypothesis test using the following hypotheses:

**H**μ = 82 (the studying technique will have no effect on the mean exam score)_{0}:**H**μ ≠ 82 (the studying technique will cause the mean exam score to be different than 82)_{A}:

This is an example of a **non-directional hypothesis** because the alternative hypothesis contains the not equal “≠” sign. The professor believes that the studying technique will influence the mean exam score, but doesn’t specify whether it will cause the mean score to increase or decrease.

**Additional Resources**

Introduction to Hypothesis Testing

Introduction to the One Sample t-test

Introduction to the Two Sample t-test

Introduction to the Paired Samples t-test