Often in statistics we want to test whether or not some assumption is true about a population parameter.

For example, we might assume that the mean weight of a certain population of turtle is 300 pounds.

To determine if this assumption is true, we’ll go out and collect a sample of turtles and weigh each of them. Using this sample data, we’ll conduct a hypothesis test.

The first step in a hypothesis test is to define the **null** and **alternative hypotheses**.

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

These two hypotheses are defined as follows:

**Null hypothesis (H _{0}):** The sample data is consistent with the prevailing belief about the population parameter.

**Alternative hypothesis (H _{A}):** The sample data suggests that the assumption made in the null hypothesis is not true. In other words, there is some non-random cause influencing the data.

**Types of Alternative Hypotheses**

There are two types of alternative hypotheses:

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 and alternative hypotheses in this case would be:

**Null hypothesis:**µ ≥ 70 inches**Alternative hypothesis:**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 and alternative hypotheses in this case would be:

**Null hypothesis:**µ = 70 inches**Alternative hypothesis:**µ ≠ 70 inches

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

**Examples of Alternative Hypotheses**

The following examples illustrate how to define the null and alternative hypotheses for different research problems.

**Example 1:** A biologist wants to test if the mean weight of a certain population of turtle is different from the widely-accepted mean weight of 300 pounds.

The null and alternative hypothesis for this research study would be:

**Null hypothesis:**µ = 300 pounds**Alternative hypothesis:**µ ≠ 300 pounds

If we reject the null hypothesis, this means we have sufficient evidence from the sample data to say that the true mean weight of this population of turtles is different from 300 pounds.

**Example 2:** An engineer wants to test whether a new battery can produce higher mean watts than the current industry standard of 50 watts.

The null and alternative hypothesis for this research study would be:

**Null hypothesis:**µ ≤ 50 watts**Alternative hypothesis:**µ > 50 watts

If we reject the null hypothesis, this means we have sufficient evidence from the sample data to say that the true mean watts produced by the new battery is greater than the current industry standard of 50 watts.

**Example 3:** A botanist wants to know if a new gardening method produces less waste than the standard gardening method that produces 20 pounds of waste.

The null and alternative hypothesis for this research study would be:

**Null hypothesis:**µ ≥ 20 pounds**Alternative hypothesis:**µ < 20 pounds

If we reject the null hypothesis, this means we have sufficient evidence from the sample data to say that the true mean weight produced by this new gardening method is less than 20 pounds.

**When to Reject the Null Hypothesis**

Whenever we conduct a hypothesis test, we use sample data to calculate a test-statistic and a corresponding p-value.

If the p-value is less than some significance level (common choices are 0.10, 0.05, and 0.01), then we reject the null hypothesis.

This means we have sufficient evidence from the sample data to say that the assumption made by the null hypothesis is not true.

If the p-value is *not* less than some significance level, then we fail to reject the null hypothesis.

This means our sample data did not provide us with evidence that the assumption made by the null hypothesis was not true.

**Additional Resource:** An Explanation of P-Values and Statistical Significance