Differences between Two-Tailed and One-Tailed Tests

This question is most relevant to hypothesis testing. For simplicity, we will use two-tailed tests for all hypothesis testing in this course, except in the figfig, where a one-tailed test is illustrated. The best way to illustrate their differences is to re-analyze the two-tailed t-test in earlier in the section as a one-tailed test.

• Null hypothesis $H_0$: $\mu_{true}= \mu_0 = 0.03$.

• Two-sided alternative hypothesis $H_a$: $\mu_{true}\not= \mu_0 = 0.03$.

• One-sided null hypothesis $H_0$: $\mu_{true}\le \mu_0 = 0.03$.

• One-sided alternative hypothesis $H_a$: $\mu_{true}\gt \mu_0 = 0.03$.

• One-sided null hypothesis $H_0$: $\mu_{true}\ge \mu_0 = 0.03$.

• One-sided alternative hypothesis $H_a$: $\mu_{true}< \mu_0 = 0.03$.

Three pairs of null and alternative hypotheses are specified. The first pair is for a two-tailed test. The second and third pairs are for one-tailed tests, with different conditions in the null hypotheses tested.

Regardless of whether a test is one-tailed or two-tailed, the test statistic remains the same, with a degree of freedom (n − 1):

$t=\frac{\bar{y}-\mu_0}{s/ \sqrt{n}}$

The choice of the acceptable Type I error threshold also remains the same: $α = 0.05$.

With the computed t-statistic and degree of freedom, the $p$ value will be identified from a t-distribution table, as before. However, the size of the $p$ value will differ, depending on the nature of the null hypothesis.

Two-tailed $p$ value: $\mu_{true}=\mu_0=0.03.$

Left-tailed $p$ value: $\mu_{true}\leq\mu_0=0.03.$

Right-tailed $p$ value: $\mu_{true}\ge\mu_0=0.03.$

The decision rule remains the same as well.

We illustrate how two-tailed and one-tailed tests can differ, using the following R code.