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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 $H_0$.

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 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 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 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 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.