A student working on a history project decided to find a 95 percent confidence interval for the difference in mean age at the time of election to office for former american presidents versus former british prime ministers. the student found the ages at the time of election to office for the members of both groups, which included all of the american presidents and all of the british prime ministers, and used a calculator to find the 95 percent confidence interval based on the t-distribution. this procedure is not appropriate in this context because

This was not the appropriate measurement because the entire populations were counted, so you would be able to just calculate the actual differences in mean and a confidence interval should not be used.

Basically if you were doing this test with the ages of all college freshman and only had a sample that was a percentage of all college freshman ever, you would need to use a confidence interval to show the range in possibilities since there is no way you could ever measure every single person.

But for US presidents and British Prime Ministers, there is a relatively small number of people so it is not difficult to actually go out and count every single one. A confidence interval shows how likely your sample matches the population - if your sample is the entire population there is no need for a confidence interval since it would be 100%.