Two-tailed vs. One-tailed Testing

Chapter 7. Hypothesis Testing: Introduction to Hypothesis Testing (p-value Approach)

Two-tailed vs. One-tailed Testing

Depending on the type of prediction that the hypotheses of a statistical test make, we say a test is either two-tailed or one-tailed.
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Two-tailed Tests

In a two-tailed or non-directional test, the hypotheses do not make a specific prediction about the direction of a treatment effect, difference, or relationship.

The hypotheses of a two-tailed #Z#-test for a population mean #\mu# are:

#H_0: \mu = \mu_0#
#H_a: \mu \neq \mu_0#

In a two-tailed test, #\mu_0# denotes the hypothesized value of the population mean under the null hypothesis.

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When there are good reasons to suspect that a treatment effect, difference, or relationship does have a specific direction, it may be beneficial to use a one-tailed test instead.
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One-tailed Tests

In a one-tailed or directional test, a prediction about the direction of a treatment effect, difference, or relationship is incorporated in the hypotheses of the test.

There are two types of one-tailed tests: left-tailed and right-tailed.

A left-tailed test should be used when the population parameter is suspected to be less than a particular value.

The hypotheses of a left-tailed #Z#-test for a population mean #\mu# are:

#H_0:\mu \geq \mu_0#
#H_a:\mu \lt \mu_0#

In a left-tailed test, #\mu_0# denotes the minimum hypothesized value of the population mean under the null hypothesis.

A right-tailed test should be used when the population parameter is suspected to be greater than a particular value.

The hypotheses of a right-tailed #Z#-test for a population mean #\mu# are:

#H_0:\mu \leq \mu_0#
#H_a:\mu \gt \mu_0#

In a right-tailed test, #\mu_0# denotes the maximum hypothesized value of the population mean under the null hypothesis.