how to find p value from t

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.

t-distrbution graph showing the right tail beyond a t value of 2.5

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.

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 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.

t distribution graph showing left tail below t value of -2.5

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 _Aif 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.

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 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 were 3. That is, the two-tailed test requires taking into account the possibility that the test statistic could fall into either tail (and 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 the 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.

t-distribution graph of two tailed probability for t values of -2.5 and 2.5

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 _Aif 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.

Now that we have reviewed the critical value and P-value approach procedures for each of 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.