Differences between Two-Tailed and One-Tailed Tests

Learn how to 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 H0H_0: μtrue=μ0=0.03\mu_{true}= \mu_0 = 0.03.

  • Two-sided alternative hypothesis HaH_a: μtrueμ0=0.03\mu_{true}\not= \mu_0 = 0.03.

  • One-sided null hypothesis H0H_0: μtrueμ0=0.03\mu_{true}\le \mu_0 = 0.03.

  • One-sided alternative hypothesis HaH_a: μtrue>μ0=0.03\mu_{true}\gt \mu_0 = 0.03.

  • One-sided null hypothesis H0H_0: μtrueμ0=0.03\mu_{true}\ge \mu_0 = 0.03.

  • One-sided alternative hypothesis HaH_a: μtrue<μ0=0.03\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=yˉμ0s/nt=\frac{\bar{y}-\mu_0}{s/ \sqrt{n}}

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

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

Two-tailed pp value: μtrue=μ0=0.03.\mu_{true}=\mu_0=0.03.

Left-tailed pp value: μtrueμ0=0.03.\mu_{true}\leq\mu_0=0.03.

Right-tailed pp value: μtrueμ0=0.03.\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.

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