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The choice of the alternative hypothesis depends on and should reflect the purpose of the hypothesis test. Three choices are possible for the alternative hypothesis.

• Two-tailed test
• Left-tailed test
• Right-tailed test

The given information states that there are three possible alternative hypothesis in a hypothesis test for a population mean are as shown below:

The population mean differs from the specified value \(\mu _{0}:\)

\(H_{a}:\mu \neq \mu ^{_{0}} \)

The population mean is greater than the specified value \(\mu _{0}:\)

\(H_{a}:\mu >\mu _{0}\)

The population mean is less than the specified value \(\mu _{0}:\)

\(H_{a}:\mu < \mu ^{_{0}}\)

Thus, the three possible alternative hypothesis are \(H_{a}:\mu \neq \mu ^{_{0}}, H_{a}:\mu >\mu _{0}\) and \(H_{a}:\mu < \mu ^{_{0}}\).