The hypothesis testing procedure described in Null and Alternative Hypothesis simply determines whether the null hypothesis should be rejected or not. Often we would like additional information.
For example, suppose the null hypothesis is that the population mean has a fixed value μ0, i.e. the null hypothesis H0: μ = μ0. Given any sample, we would like to use the data in the sample to calculate an interval (called a confidence interval) corresponding to that sample such that 95% of such samples will produce a confidence interval which contains the the population mean μ (where α = .05, and so 95% = 1 – α); i.e. we are 95% confident that a < μ < b where a and b are the end points of the interval. Furthermore, if a < μ0 < b, then we can’t reject the null hypothesis, while if μ0 ≥ b or μ0 ≤ a, then we can reject the null hypothesis.