When learning about testing hypothesis \(H_{0}\) and \(H_{1}\) in elementary statistics courses, students are usually tasked with computing a “p-value” on their TI-83 calculators and “rejecting the null hypothesis if the p-value is less than 0.05”. As a statistics instructor myself, I’ve seen many students know how to decide upon a rejection by memorizing the “p-value rule” without being taught why such a rule exists.
One of the first definitions a student learns in an elementary statistics course is that an event is described as unusual if the probability of occurring is less than or equal to 0.05. The same rule here applies to hypothesis testing, which is why \(\alpha\) is often given to be 0.05.
However, a definition that students often do not learn in an elementary statistics course is the definition of the p-value. A p-value is defined as the probability of getting a value for that test statistic as extreme as or more extreme than what was actually observed given that \(H_0\) is true. From another perspective, the p-value is written as the probability of getting results as “unusual” as ours, if \(H_0\) is actually correct. A third way of looking at p-value is given as probability of a type I error if \(H_0\) is correct, or \(P(\)Type I error\()\).
Let’s consider a quick hypothesis test problem that students might learn in a statistics course. Suppose General Motors claims the average gas mileage of 2019 Cadillac CTS-V’s is given 14 MPG. We believe the average gas mileage (\(\mu\)) is actually less than 14, so we sample 25 Cadillac CTS-V’s and find out that they have an average gas mileage of 12.1 MPG(\(\bar{x}\)) with standard deviation 5.5 MPG(\(\sigma\)). Use \(\alpha = 0.05\). Assume conditions are met since this is all hypothetical.
Here, our hypotheses are:
\[H_0: \mu = 14\] \[H_1: \mu < 14\] A quick calculation would show that our t-statistic is given by \(-1.727273\) and our p-value is given as \(0.04848\).
We can see that our p-value \(0.04848 < 0.05\), which means that there is approximately a 4.85% chance of obtaining the results we did if \(H_0\) is correct. From the first week of our elementary statistics course, we can recall that this would be an unusual event, so the null hypothesis must not be correct.