What does it mean to fail to reject the null hypothesis?

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.

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.

The null hypothesis serves as the reference claim that includes the boundary value. In the example, it represents the status quo or the specific departure being tested against the alternative hypothesis.

Conditions: The population mean is 23.; The null hypothesis includes the boundary value.