You want to determine whether a new gasoline additive has an effect on gas mileage. If the known gas mileage for this specific class of cars is 25 miles per gallon (mpg), the hypotheses for this study are H_{0}: μ = 25 and H_{A}: μ ≠ 25.

You test 35 cars and discover that the miles per gallon the cars get ranges from 14.4 to 28.8. After putting the data in the column MPG, you perform Minitab’s t-test (the menu command TTEST) and get these results: ### One-Sample T: C1

### T-Value P-Value
-2.33 0.026

, or the session command Descriptive Statistics
N Mean StDev SE Mean 95% CI for μ
35 23.657 3.416 0.577 (22.484, 24.831)
μ: mean of C1

Test
Null hypothesis H₀: μ = 25
Alternative hypothesis H₁: μ ≠ 25

The results show that the mean of the 35-car sample is 23.657. But the mean miles per gallon of all cars of this type (μ) might still be 25. You need to know whether there is enough sample evidence to reject H_{0}. The most common way is to compare the p-value to the significance level, α (alpha). α is the probability of rejecting H_{0} when H_{0} is true. In this case, that’s the probability of concluding that the population mean is *not* 25 mpg, when in fact it is.

The p-value is a measure of the strength of the evidence in your data against H_{0}. Usually, the smaller the p-value, the stronger the sample evidence is for rejecting H_{0}. More specifically, the p-value is the smallest value of α that results in the rejection of H_{0}. For any value of α > p-value, you fail to reject H_{0}, and for any value of α p-value, you reject H_{0}.

In our t-test example, the test statistic is a function of the mean, and the p-value is .026. This indicates that 2.6% of the samples of size 35, drawn from the population where μ = 25, will produce a mean that provides as strong (or stronger) evidence as the current sample that μ is not equal to 25. Ask yourself which is more likely: that μ = 25 and you just happened to select a very unusual sample; or that μ is not equal to 25?

The p-value is traditionally compared to α values of less than .05 or .01, depending on the field of study. Check journal topics in your field for acceptable values.

In our example, let’s assume a .05 value of α. The p-value of .026 indicates that the mean miles per gallon of all cars of this type (not only the mean of the 35 cars in the study) is probably not equal to 25. A more statistically correct way to state this is “at a significance level of .05, the mean miles per gallon seems to be significantly different from 25.”

Using p-values, then, is simple if you know two key facts: the values of α that are acceptable in your field, and the null and alternative hypotheses for the tests you are using.