Welcome to our guide on calculating p-values. The p-value is a fundamental concept in statistical hypothesis testing. It represents the probability of observing data as extreme as, or more extreme than, what we actually collected, assuming our null hypothesis is true. In this visualization, we see a normal distribution with a test statistic at 1.96, and the shaded red area represents the p-value.
The first step in calculating a p-value is to formulate your hypotheses. You need a null hypothesis, which typically states there is no effect or no difference, and an alternative hypothesis, which states there is an effect. For example, when testing if a coin is fair, the null hypothesis states that the probability of heads equals 0.5, while the alternative hypothesis states it does not equal 0.5.
The second step is to calculate the test statistic from your sample data. The formula depends on which statistical test you're using. For a z-test with known population standard deviation, we use the z-formula. For a t-test with unknown population standard deviation, we use the t-formula. In our example, with a sample mean of 52, null hypothesis mean of 50, sample standard deviation of 8, and sample size of 25, we calculate t equals 1.25.
The third step is to find the p-value using your test statistic and the appropriate probability distribution. For a one-tailed test, the p-value is the probability of getting a test statistic greater than or equal to the absolute value of your calculated statistic. For a two-tailed test, you multiply this probability by two. In our example with t equals 1.25 and 24 degrees of freedom, the two-tailed p-value is 2 times 0.111, which equals 0.222.
To summarize what we've learned about calculating p-values: The p-value measures the probability of observing extreme data under the null hypothesis. First, formulate clear hypotheses. Second, calculate the appropriate test statistic from your sample data. Third, find the p-value using the correct probability distribution. Finally, compare the p-value to your significance level to make a statistical decision.