How to Calculate a P-Value When Testing a Null Hypothesis

A p-value is a probability associated with your critical value. The critical value depends on the probability you are allowing for a Type I error. It measures the chance of getting results at least as strong as yours if the claim (H₀) were true.

The following figure shows the locations of a test statistic and their corresponding conclusions.

Decisions for Ha: not-equal-to

To find the p-value for your test statistic:

1. Look up your test statistic on the appropriate distribution — in this case, on the standard normal (Z-) distribution (see the following Z-tables).

   
   

2. Find the probability that Z is beyond (more extreme than) your test statistic:

   1. If H_a contains a less-than alternative, find the probability that Z is less than your test statistic (that is, look up your test statistic on the Z-table and find its corresponding probability). This is the p-value. (Note: In this case, your test statistic is usually negative.)

   2. If H_a contains a greater-than alternative, find the probability that Z is greater than your test statistic (look up your test statistic on the Z-table, find its corresponding probability, and subtract it from one). The result is your p-value. (Note: In this case, your test statistic is usually positive.)

   3. If H_a contains a not-equal-to alternative, find the probability that Z is beyond your test statistic and double it. There are two cases:

      If your test statistic is negative, first find the probability that Z is less than your test statistic (look up your test statistic on the Z-table and find its corresponding probability). Then double this probability to get the p-value.

      If your test statistic is positive, first find the probability that Z is greater than your test statistic (look up your test statistic on the Z-table, find its corresponding probability, and subtract it from one). Then double this result to get the p-value.

These calculations give you a test statistic (standard score) of –0.05 divided by 0.04 = –1.25. This tells you that your sample results and the population claim in H₀ are 1.25 standard errors apart; in particular, your sample results are 1.25 standard errors below the claim.

When testing H₀: p = 0.25 versus H_a: p < 0.25, you find that the p-value of -1.25 by finding the probability that Z is less than -1.25. When you look this number up on the above Z-table, you find a probability of 0.1056 of Z being less than this value.

Note: If you had been testing the two-sided alternative,

the p-value would be 2 ∗ 0.1056, or 0.2112.

If the results are likely to have occurred under the claim, then you fail to reject H₀ (like a jury decides not guilty). If the results are unlikely to have occurred under the claim, then you reject H₀ (like a jury decides guilty).