We define a simple random sample of size n as a sample chosen by the method in which each collection of n population items is equally likely to be selected.

Question

Consider the population of all senators in the senate today. We select a sample of two senators by randomly selecting a state and then taking two senators representing the state selected. Is this a simple random sample of size 2?

Answer 1

No, the method described does not constitute a simple random sample of size 2 from the population of all senators in the Senate. In a simple random sample, each possible collection of 2 senators must have an equal chance of being selected.

In your described method, the process involves first randomly selecting a state and then taking the two senators representing that state. This means that the populations of senators are not equally represented, as states vary in size and some states have more representation than others (e.g., California has two senators, while Wyoming also has two, but taking two senators from a less populous state limits the combinations available). Consequently, not every possible pair of senators from the entire Senate has an equal chance of selection, thus violating the principle of simple random sampling.