the moment the fuel light in my car goes on, I will find a closest gas station to fill up the tank. But the # of gallons needed to fill up are all over the place: 1) how can we verify it is actually 1 gallon when the pump says it pumps 1 gallon of gas into your car, 2) if we know some gas stations are short selling us, how to find out what the true volume of gas is.

These 2 probS are known as hypothesis testing (testing the claim, i.e., the hypothesis, that a supposed gallon is indeed one gallon), and estimation (estimating the true volume) in statistical inference.

Since the volume is measured, i.e., a continuous variable, you know if you take a measure of the volume of a supposed gallon of gas pumped out the of machine, it will never be exactly 1.0000000... gallon.

So if it turns out to be 0.9823, can you say they are cheating?

You might say I will take 30 measures and look at their mean, but again if it turns out to be 0.9912? can you say they are cheating?

1. Outline your thoughts and share.The statistical inference on modern stat textbooks are the results of two statisticians's answer (Fisher's p-value and Neyman's critical values) to these questions, and they are not even in full agreement.