Are the Observations in A Random Sample Independent?

Hi, guys.

I am learning the t-test on my own.

The assumptions of t-test have been confusing to me.

The assumptions included "A random sample is used" and "The random sample is made up of independent observations", so I wondered the observations in a random sample may not be independent given the assumptions.

However, I really cannot figure out an example to illustrate my idea.

Could you please give me an example? Thanks.

assume that you want to find out if the average amount of animals differs between two different habitats. Thus, you use a grid in each habitat and and you randomly choose 100 grid cells (e.g. out of 10.000) from each type of landscape. Thus, you have two random samples and you can compare them via T-test. Now you can nevertheless have the case that these "observations" (which means in this case number of animals per grid cell) are not independent: (1) you can have a dependency in space: The closer two cell are, the more likely is that they have similar animal numbers. This is called a "spatial autocorrelation" and the assumption of independence would be violated. (2) you can observe the same effect regarding time: The shorter the time is between the counts in your grid cells, the more likely is could that they have similar values (imaginge that you have evenly spread animals but the density chaning in time, e.g. due to a dayly walk to some feeding grounds outside). This is called "temporal autocorrelation" and is a violation of independence as well. You see: A randiom sample does not per se prevent such dependencies. That is the reason why you always should check this, even if you have a random design.


TS Contributor
I think the independence between the two samples is important but the independence of the data points within each sample much less so.