*59*

AÂ hypothesis test is used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using 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}):

If the p-value of the hypothesis test is less than some significance level (e.g. Î± = .05), then **we reject the null hypothesis**.

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

When writing the conclusion of a hypothesis test, we typically include:

- Whether we reject or fail to reject the null hypothesis.
- The significance level.
- A short explanation in the context of the hypothesis test.

For example, we would write:

We

reject the null hypothesisat the 5% significance level.Â

There is sufficient evidence to support the claim thatâ€¦

Or, we would write:

We

fail to reject the null hypothesisat the 5% significance level.Â

There is not sufficient evidence to support the claim thatâ€¦

The following examples show how to write a hypothesis test conclusion in both scenarios.

**Example 1: Reject the Null Hypothesis Conclusion**

Suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.

She then performs a hypothesis test at a 5% significance level using the following hypotheses:

**H**Î¼ = 20 inches (the fertilizer will have no effect on the mean plant growth)_{0}:**H**Î¼ > 20 inches (the fertilizer will cause mean plant growth to increase)_{A}:

Suppose the p-value of the test turns out to be 0.002.

Here is how she would report the results of the hypothesis test:

We

reject the null hypothesisat the 5% significance level.Â

There is sufficient evidence to support the claim that this particular fertilizer causes plants to grow more during a one-month period than they normally do.

**Example 2: Fail to Reject the Null Hypothesis Conclusion**

Suppose the manager of a manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, he measures the mean number of defective widgets produced before and after using the new method for one month.

He performs a hypothesis test at a 10% significance level using the following hypotheses:

**H**Î¼_{0}:_{after}= Î¼_{before}(the mean number of defective widgets is the same before and after using the new method)**H**Î¼_{A}:_{after}â‰ Î¼_{before}(the mean number of defective widgets produced is different before and after using the new method)

Suppose the p-value of the test turns out to be 0.27.

Here is how he would report the results of the hypothesis test:

We

fail to reject the null hypothesisat the 10% significance level.Â

There is not sufficient evidence to support the claim that the new method leads to a change in the number of defective widgets produced per month.

**Additional Resources**

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing

4 Examples of Hypothesis Testing in Real Life

How to Write a Null Hypothesis