Performing p-value Hypothesis Tests

This blog is aimed at students and focuses on how to perform a hypothesis test using the p-value approach.

What is a p-value?

The p-value is the probability of getting a sample statistic as extreme or more extreme than the one obtained if the null hypothesis is true.

For example, if you were testing the claim that a population mean was less that 25 and your sample of 144 observations had a mean of 21 with a standard deviation of 2, the p-value is the probability that a sample mean is less than 21 under those conditions.

The lower this probability is, the less likely it is that the null hypothesis is true.

Step 1 – State the null and alternate hypotheses.

• List the null hypothesis $H_0$ and the alternate hypothesis $H_1$.
• The null hypothesis is assumed to be true, and we look for evidence (a low p-value) that the alternate hypothesis is true.
• The null hypothesis always contains equality, and the alternate hypothesis will always contain one of these three symbols: $<, >, \ne$.

Step 2 – Establish a level of significance.

• The level of significance, $\alpha$, is the equivalent of reasonable doubt in a jury trial.
• If the p-value is less than $\alpha$, then we consider this to be strong evidence that the null hypothesis is false (Reject $H_0$) and the alternate hypothesis is true (Support $H_1$).
• In homework problems the level of significance is provided, but when performing your own hypothesis test choose a level of significance that corresponds to the consequences of making a wrong decision. Choosing $\alpha = 0.05$ is pretty standard, and you have a 5% chance of rejecting a null hypothesis that is actually true. If the consequences of making such an error are high (someone could get sick, you could be sued, …), consider using $\alpha = 0.01$.

Step 3 – Choose a test statistic.

• The test statistic is a formula we use to calculate the p-value for a hypothesis test.
• The formula we use depends on the test we are performing. For instance, if you were performing a 1-mean t test, the test statistic would be:
$t = \frac {\bar{x} - \mu}{s/\sqrt{n}}$
• In a later blog I will share advice on how to select the correct test statistic.

Step 4 – Establish the decision rule.

• The decision rule is a rule that determines when the null hypothesis will be rejected.
• When using the p-value approach, the decision rule is always “Reject $H_0$ if p-value < $\alpha$.”

Step 5 – Calculations, Decision about $H_0$, Conclusion

• By Hand
To calculate the p-value, begin by calculating the test statistic.
Use that test statistic and the appropriate probability table, along with the direction of the test suggested by $H_1$, to calculate the p-value.
• By Technology
Whether you are using StatCrunch, Excel, Minitab, the TI-83 or TI-84 calculator, or some other statistical package, the p-value will be calculated for you. (In today’s world, you should be using technology to find p-values!)
• If the p-value is less than the level of significance, we reject $H_0$ and support $H_1$. Otherwise we fail to reject $H_0$ (we cannot conclude $H_0$ is false) and we fail to support $H_1$ (we cannot conclude that $H_1$ is true).
• Finish by writing a conclusion about the claim you were testing.

Summary

I hope this is helpful to you. In a later blog I will go over how to choose the appropriate test (and test statistic). I will also go over how to perform hypothesis tests using StatCrunch. I’ll conclude by saying that StatCrunch is finally allowing me to teach this course the way it should be taught. If you want to see what StatCrunch is all about for yourself, check out their web site at http://StatCrunch.com .

-George

I am a math instructor at College of the Sequoias in Visalia, CA. Each Wednesday I post an article related to General Teaching on my blog. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website – http://georgewoodbury.com