In statistics, we use hypothesis tests to determine whether some claim about a population parameter is true or not.

Whenever we perform a hypothesis test, we always write a **null hypothesis** and an **alternative hypothesis**, which take the following forms:

**H _{0}** (Null Hypothesis): Population parameter = ≤, ≥ some value

**H _{A}** (Alternative Hypothesis): Population parameter <, >, ≠ some value

There are three different types of hypothesis tests:

- Two-tailed test: The alternative hypothesis contains the “≠” sign
- Left-tailed test: The alternative hypothesis contains the “<” sign
- Right-tailed test: The alternative hypothesis contains the “>” sign

Notice that we only have to look at the sign in the alternative hypothesis to determine the type of hypothesis test.

Left-tailed test:The alternative hypothesis contains the “<” sign

Right-tailed test:The alternative hypothesis contains the “>” sign

The following examples show how to identify left-tailed and right-tailed tests in practice.

**Example: Left-Tailed Test**

Suppose it’s assumed that the average weight of a certain widget produced at a factory is 20 grams. However, one inspector believes the true average weight is less than 20 grams.

To test this, he weighs a simple random sample of 20 widgets and obtains the following information:

- n =
**20**widgets - x =
**19.8**grams - s =
**3.1**grams

He then performs a hypothesis test using the following null and alternative hypotheses:

**H _{0}** (Null Hypothesis): μ ≥ 20 grams

**H _{A}** (Alternative Hypothesis): μ < 20 grams

The test statistic is calculated as:

*t*= (x – µ) / (s/√n)*t*= (19.8-20) / (3.1/√20)*t*= -.2885

According to the t-Distribution table, the t critical value at α = .05 and n-1 = 19 degrees of freedom is –**1.729**.

Since the test statistic is not less than this value, the inspector fails to reject the null hypothesis. He does not have sufficient evidence to say that the true mean weight of widgets produced at this factory is less than 20 grams.

**Example: Right-Tailed Test**

Suppose it’s assumed that the average height of a certain species of plant is 10 inches tall. However, one botanist claims the true average height is greater than 10 inches.

To test this claim, she goes out and measures the height of a simple random sample of 15 plants and obtains the following information:

- n =
**15**plants - x =
**11.4**inches - s =
**2.5**inches

She then performs a hypothesis test using the following null and alternative hypotheses:

**H _{0}** (Null Hypothesis): μ ≤ 10 inches

**H _{A}** (Alternative Hypothesis): μ > 10 inches

The test statistic is calculated as:

*t*= (x – µ) / (s/√n)*t*= (11.4-10) / (2.5/√15)*t*= 2.1689

According to the t-Distribution table, the t critical value at α = .05 and n-1 = 14 degrees of freedom is **1.761**.

Since the test statistic is greater than this value, the botanist can reject the null hypothesis. She has sufficient evidence to say that the true mean height for this species of plant is greater than 10 inches.

**Additional Resources**

How to Read the t-Distribution Table

One Sample t-test Calculator

Two Sample t-test Calculator