What’s the difference in practical terms between calculating the probability of having a sample mean smaller than x and the probability of an individual of a population having a value smaller than x?
For example: assume I am studying my travel time from home to office. I take a sample of 30 values. I can calculate: A) the probability of having a mean smaller than 60 minutes. B) The probability of a single trip taken by chance to last up to 60 minutes.
Let’s say I want to analyze the risk of arriving in the office after 60 minutes. Which approach would better represent this point? I’d like to understand the differences of these 2 approaches in real life decision making.
Generally, the "mean" is not a random variable. However, when speaking of the "mean" you probably mean the "sample mean". The latter is the mean you observe after seeing 30 samples. (A) The event of having a sample mean < 60 minutes is the event of on average arriving within 60 minutes. (B) The event of having a single trip < 60 minutes is the event of on arriving on that particular day within 60 minutes.