However, since the null hypothesis would be only one of an infinite number of values in the confidence interval, accepting the null hypothesis is not justified.

There are many arguments against accepting the null hypothesis when it is not rejected. The null hypothesis is usually a hypothesis of no difference. Thus null hypotheses such as:

μ

π

in which the hypothesized value is zero are most common. When the hypothesized value is zero then there is a simple relationship between hypothesis testing and confidence intervals:

If the interval contains zero then the null hypothesis cannot be rejected at the stated level of confidence. If the interval does not contain zero then the null hypothesis can be rejected.This is just a special case of the general rule stating that the null hypothesis can be rejected if the interval does not contain the hypothesized value of the parameter and cannot be rejected if the interval contains the hypothesized value.