Simple Random Sampling (SRS) assumes that all individuals in the population have the same probability of being selected. The sample mean \(\bar{X}\) is defined as:

\[ \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i, \]

where \(X_i\) represents the salary of the \(i\)-th professor in the sample, and \(n\) is the sample size.

In Cluster Sampling, the population is divided into groups (universities), and some of these clusters are randomly selected, after which all individuals within them are surveyed. The cluster sample mean can be expressed as:

\[ \bar{X}_{clu} = \frac{1}{n_{clu}} \sum_{k \in S} \bar{X}_k, \]

where \(\bar{X}_k\) is the mean of the variable of interest (salary) in the \(k\)-th cluster, and \(n_{clu}\) is the number of clusters selected in the sample.