Statsland is a magical world that exists only in (certain) Statistics textbooks. In Statsland, statistics is easy. We can invoke Central Limit theorem and use the normal distribution when n is larger than 30. In Statsland we either know or can easily determine the correct distribution. In Statsland 95% confidence intervals have a 95% chance of containing the real value. But we don’t live in Statsland.
The point of doing statistics is that it would be too difficult (or impossible) to find the true value of a population. You aren’t likely to find the exact value, but you can be pretty close. In a statistics textbook problem, you probably have enough information to do a good job of estimating the desired value. But in applied statistics you may not have as much information. If you know the mean and standard deviation of a population you do not need to do much (if any) statistics. Any time you have to estimate or substitute information, your model will not perform as well as a theoretically perfect model.
Statistics never was and never will be an exact science. In most cases, your model will be wrong. There are no perfect answers. Your confidence intervals will rarely perform as they theoretically should. The requisite sample size to invoke Central Limit Theorem is not clear cut. Your approach should vary on the individual problem. There is no universal formula to examine data. Applied Statistics should be flexible and instead of rigid. The world is not a statistics textbook problem, and should never be treated as such.
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I’m going to comment here in case anyone ever reads this and thinks I don’t know what I’m talking about. Yes I know the 95% of being in a confidence interval is an asymptotic result and n=30 is an arbitrary cutoff. My point is there is a tendency to put meaning on statistics when it is not there.