You must report the probability that a sum of independent observations lands more than five standard deviations above its mean, where is a few hundred and you know only the mean, the standard deviation and the fact that each observation is bounded. Section 12.4 offers a shape: the sum is approximately normal, so the answer is one number in the far tail of the bell. Section 12.5 offers bounds that need no shape and give an answer many orders of magnitude larger. Section 12.5 also shows that Chebyshev's bound is silent when it exceeds one, and that its linear decay is far slower than the exponential decay of a bound that uses boundedness. The field has no general rule that says, for a given and a given source, that the normal number is accurate enough to report; the size of the error in the normal approximation depends on the source, and the far tail is where the approximation is weakest.
Take a position on this question: what number do you report for this tail probability, what do you assume to get it, and what does your report promise?
In your answer: state the procedure that produces the number; say which assumption carries the normal approximation and how you would test it with the data you have; say what the reported number covers and what it leaves out; and name one way your procedure could mislead you even when every step is followed.
Rubric. Tick each criterion your answer meets, then save the self-assessment. The platform does not grade it.
Worked solution
There is no single correct answer, so this is a model of the shape a complete answer takes, not a key. A level-3 answer begins by fixing the event and the two candidate numbers. The normal candidate comes from the bell: the sum sits five standard deviations above its mean, so the tail probability is about on one side. The bound candidate comes from Chebyshev, which at five standard deviations gives on one side. The gap between them, about five orders of magnitude, is the first fact the answer reports, because it shows how much the shape is doing.
The answer then says what the normal number assumes: that the standardised sum is close enough to normal at this for the far tail, which is a stronger demand than closeness near the centre. It proposes a check that does not use the tail being estimated, such as comparing the lopsidedness and the tail weight of the observations with the values a normal source would give, or splitting the data and comparing the empirical tail of the sum on one part with the prediction on the other. It reports the bound as the fallback and states which number it would defend in public.
The report promises the sampling behaviour of the chosen number under the stated model over repeated samples of this size. It does not promise that the source is normal, that the observations are independent, or that the far tail is not dominated by a few large values. The answer gives the cost: the check uses rows that could have gone into the estimate, and a check that passes at the centre says little about the tail. It names a failure: a heavy-tailed source can pass a moment check at in the hundreds and still put far more probability beyond five standard deviations than the bell does, so the normal number is wrong in the direction that matters for a risk report. A strong answer says what would change its mind, such as a larger , a bound that uses the known range of the observations, or evidence that the largest observations are not the ones driving the sum.