How does a one-tailed test allocate the rejection region?

In a symmetric two-tailed construction, the significance level $\alpha$ is split equally between the two tails. Each tail receives $\alpha/2$.

Conditions: The test is a symmetric two-tailed test.; The significance level is $\alpha$.

The form of the alternative hypothesis dictates the tail(s) of the test. $H_1:\mu\ne23$ gives a two-tailed test, $H_1:\mu<23$ a left-tailed test, and $H_1:\mu>23$ a right-tailed test.

Conditions: The null hypothesis is based on a population mean of 23.; The alternative hypothesis is formulated before seeing the data.

Failing to reject the null hypothesis means that the sample provides insufficient evidence against it. It is not a proof that the null is true; it simply indicates that the data do not contradict the null hypothesis strongly enough to warrant rejection.

Conditions: A hypothesis test has been conducted.; The decision is to fail to reject $H_0$.

The significance level $\alpha$ controls the chance of rejecting a true null hypothesis. It defines the probability of making a Type I error, which is rejecting $H_0$ when it is actually true.

Conditions: The test is conducted under the null model.; $\alpha$ is the pre-specified significance level.

The null hypothesis includes the boundary value. In the example with a population mean of 23, the null hypothesis is centered around or includes this specific value as the reference claim against which the alternative hypothesis tests for departure.

Conditions: The example uses a population mean of 23.; The null hypothesis is formulated to include this value.