I have recently been learning statistics from the ground up (for context I am an engineer by training with no formal education in statistics/probability).
I have seen various hypothesis tests: z-test, t-test, 2-sample z or t test, paired differences test, etc. In the examples I have seen, these usually involve 1 or 2 samples: for example, a t-test was used when analysing the lead concentration in a small number of water samples, comparing it to some expected safety baseline. Or, in the case of the 2-sample z-test, 2 samples of the voting population were taken 1 month apart, and a 2 sample z test was applied to see if the difference was statistically significant. My understanding of the hypothesis test (at the 95% significance level for example) is that we want to be 95% confidence that our results did not come from chance error in sampling i.e. just random fluctuations in the population.
So, my question: does it make sense to apply a hypothesis test when we believe we have the entire population data?
For example, say we are considering player performances in soccer. We have 2 seasons, 2021/22 and 2022/23. We take a player, say Erling Haaland, and compute his mean number of goals per match (38 matches in each season):
- Does it make sense in this scenario to apply a 2 sample t-test to the mean number of goals per match to see if the result is statistically significant (i.e. is there a statistically significant increase or decrease in performance between seasons)? Since we have all the data from his 38 matches in each season, we can easily see whether his mean goals per match has changed or not without any kind of statistical test.
- Or can we look at it through another lens: where Haaland has an (unknown) underlying data generating histogram for goals per match each season, and the 38 matches are sampled from this true, unknown probability histogram. Therefore, if we do a 2 sample t-test on his mean goals per match each season, we are trying to determine if the mean of these underlying data generating histograms are different?
Apologies for the long post and any answers or feedback is appreciated, thanks.
