How to find the p-value for one-tailed and two-tailed tests

Once you’ve calculated a test statistic, the next step is to ask how extreme it would be if the null hypothesis were true. We’ll use three z-scores to see which area to count and how that area helps us make a decision.

Find the area for results at least as extreme

The p-value is the probability, assuming the null hypothesis $H_{0}$ is true, of getting a test statistic at least as extreme as the one we observed, in the direction or directions given by the alternative hypothesis. In a lower-tailed test, count smaller values. In an upper-tailed test, count larger values. For a two-tailed z-test, count values at least as far from zero in either direction.

Find a lower-tailed p-value

If the observed statistic is $z = - 1.46$ , a left-area z-table gives $0.0721$ . You can find it in the $- 1.4$ row and the $.06$ column. The table already gives the area we need, so $p \approx 0.0721$ .

The p-value is the area to the left of −1.46.

Find an upper-tailed p-value

If the observed statistic is $z = 1.46$ , the table gives $0.9279$ to the left. The two areas add up to $1$ , so subtract the left area to get the right area.

p \approx 1 - 0.9279 = 0.0721

The p-value is the area to the right of 1.46.

Find a two-tailed p-value

For $z = 1.23$ , the z-table gives $0.8907$ to the left. The right tail starts at $1.23$ . Its matching left tail ends at $- 1.23$ . Because the normal curve is symmetric, those two areas are equal.

\begin{array}{l} 1 - 0.8907 = 0.1093 \\ p \approx 2 (0.1093) = 0.2186 \end{array}

The two shaded tails have equal area, so we count both.

The same tail logic works for the symmetric $t$ distribution. Use the appropriate degrees of freedom and check whether your table reports a left area, a right-tail area, or both tails together.

Compare the p-value with the significance level

Choose the significance level $α$ before looking at the results. It sets the allowable Type I error rate, the chance of rejecting a true null hypothesis under the test model. If $p \leq α$ , reject $H_{0}$ . If $p > α$ , fail to reject $H_{0}$ .

At $α = 0.10$ , both one-tailed examples are significant because $0.0721 \leq 0.10$ . The two-tailed example fails to reject because $0.2186 > 0.10$ . At $α = 0.05$ or $α = 0.01$ , all three fail to reject.

Use a critical value to make the same decision

In an upper-tailed z-test at $α = 0.05$ , the critical value is about $1.645$ . It leaves $0.05$ to its right. The observed $z = 1.46$ falls below that cutoff, so we fail to reject $H_{0}$ . Here $α$ sets the rejection region. The p-value is the area measured from the observed statistic.

Learn mathKrista KingSeptember 13, 2020math, learn online, online course, online math, probability and statistics, probability and stats, probability, statistics, stats, p-value, rejecting the null, one-tail test, two-tail test, upper-tail test, lower-tail test