The *P*-value approach involves determining "likely" or "unlikely" by determining the probability — assuming the null hypothesis was true — of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed. If the *P*-value is small, say less than (or equal to) \(\alpha\), then it is "unlikely." And, if the *P*-value is large, say more than \(\alpha\), then it is "likely."

If the *P*-value is less than (or equal to) \(\alpha\), then the null hypothesis is rejected in favor of the alternative hypothesis. And, if the *P*-value is greater than \(\alpha\), then the null hypothesis is not rejected.

Specifically, the four steps involved in using the *P*-value approach to conducting any hypothesis test are:

- Specify the null and alternative hypotheses.
- Using the sample data and assuming the null hypothesis is true, calculate the value of the test statistic. Again, to conduct the hypothesis test for the population mean
*μ*, we use the*t*-statistic \(t^*=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) which follows a*t*-distribution with*n*- 1 degrees of freedom. - Using the known distribution of the test statistic, calculate the
**P****-value**: "If the null hypothesis is true, what is the probability that we'd observe a more extreme test statistic in the direction of the alternative hypothesis than we did?" (Note how this question is equivalent to the question answered in criminal trials: "If the defendant is innocent, what is the chance that we'd observe such extreme criminal evidence?") - Set the significance level, \(\alpha\), the probability of making a Type I error to be small — 0.01, 0.05, or 0.10. Compare the
*P*-value to \(\alpha\). If the*P*-value is less than (or equal to) \(\alpha\), reject the null hypothesis in favor of the alternative hypothesis. If the*P*-value is greater than \(\alpha\), do not reject the null hypothesis.

Example S.3.2.1

### Mean GPA Section

In our example concerning the mean grade point average, suppose that our random sample of *n* = 15 students majoring in mathematics yields a test statistic *t** equaling 2.5. Since *n* = 15, our test statistic *t** has *n *- 1 = 14 degrees of freedom. Also, suppose we set our significance level α at 0.05 so that we have only a 5% chance of making a Type I error.

### Right Tailed

The *P*-value for conducting the **right-tailed** test *H*_{0} : *μ* = 3 versus *H*_{A} : *μ* > 3 is the probability that we would observe a test statistic greater than *t** = 2.5 if the population mean \(\mu\) really were 3. Recall that probability equals the area under the probability curve. The *P*-value is therefore the area under a *t*_{n - 1} = *t*_{14} curve and to the *right *of the test statistic *t** = 2.5. It can be shown using statistical software that the *P*-value is 0.0127. The graph depicts this visually.

The *P*-value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic *t** in the direction of *H*_{A} if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the *P*-value, 0.0127, is less than\(\alpha\) = 0.05, we reject the null hypothesis *H*_{0} : *μ* = 3 in favor of the alternative hypothesis *H*_{A} : *μ* > 3.

Note that we would not reject *H*_{0} : *μ* = 3 in favor of *H*_{A} : *μ* > 3 if we lowered our willingness to make a Type I error to\(\alpha\) = 0.01 instead, as the *P*-value, 0.0127, is then greater than\(\alpha\) = 0.01.

### Left Tailed

In our example concerning the mean grade point average, suppose that our random sample of *n* = 15 students majoring in mathematics yields a test statistic *t** instead of equaling -2.5. The *P*-value for conducting the **left-tailed** test *H*_{0} : *μ* = 3 versus *H*_{A} : *μ* < 3 is the probability that we would observe a test statistic less than *t** = -2.5 if the population mean *μ* really were 3. The *P*-value is therefore the area under a *t*_{n - 1} = *t*_{14} curve and to the *left *of the test statistic t* = -2.5. It can be shown using statistical software that the *P*-value is 0.0127. The graph depicts this visually.

The *P*-value, 0.0127, tells us it is "unlikely" that we would observe such an extreme test statistic *t** in the direction of *H*_{A }if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the *P*-value, 0.0127, is less than α = 0.05, we reject the null hypothesis *H*_{0} : *μ* = 3 in favor of the alternative hypothesis *H*_{A} : *μ* < 3.

Note that we would not reject *H*_{0} : *μ* = 3 in favor of *H*_{A} : *μ* < 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the *P*-value, 0.0127, is then greater than\(\alpha\) = 0.01.

### Two-Tailed

In our example concerning the mean grade point average, suppose again that our random sample of *n* = 15 students majoring in mathematics yields a test statistic *t** instead of equaling -2.5. The *P*-value for conducting the **two-tailed** test *H*_{0} : *μ* = 3 versus *H*_{A} : *μ* ≠ 3 is the probability that we would observe a test statistic less than -2.5 or greater than 2.5 if the population mean *μ* really was 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (hence the name "two-tailed" test). The *P*-value is, therefore, the area under a *t*_{n - 1} = *t*_{14} curve to the *left *of -2.5 and to the *right* of 2.5. It can be shown using statistical software that the *P*-value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually.

Note that the *P*-value for a two-tailed test is always two times the *P*-value for either of the one-tailed tests. The *P*-value, 0.0254, tells us it is "unlikely" that we would observe such an extreme test statistic *t** in the direction of *H*_{A }if the null hypothesis were true. Therefore, our initial assumption that the null hypothesis is true must be incorrect. That is, since the *P*-value, 0.0254, is less than α = 0.05, we reject the null hypothesis *H*_{0} : *μ* = 3 in favor of the alternative hypothesis *H*_{A} : *μ* ≠ 3.

Note that we would not reject *H*_{0 }: *μ* = 3 in favor of *H*_{A} : *μ* ≠ 3 if we lowered our willingness to make a Type I error to α = 0.01 instead, as the *P*-value, 0.0254, is then greater than\(\alpha\) = 0.01.

Now that we have reviewed the critical value and *P*-value approach procedures for each of the three possible hypotheses, let's look at three new examples — one of a right-tailed test, one of a left-tailed test, and one of a two-tailed test.

The good news is that, whenever possible, we will take advantage of the test statistics and *P*-values reported in statistical software, such as Minitab, to conduct our hypothesis tests in this course.