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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:
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.
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.
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.
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.
