Exercise 3: Why the t-distribution Needs Heavier Tails — Possible Solution
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WHY HEAVIER TAILS ARE NEEDED WHEN sigma IS UNKNOWN
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A z-test assumes the true population standard deviation, sigma, is
already known with certainty - all of the uncertainty in the test
comes from a single source, ordinary sampling error in the mean
itself. A t-test, per this chapter's own explanation, instead has to
ESTIMATE sigma from the very same limited sample using s, which
introduces a second, separate source of uncertainty on top of the
first: the estimate s could itself be a bit too high or too low,
purely by chance, especially with a small sample. The t-distribution's
heavier tails exist specifically to account for that extra layer of
uncertainty - they assign somewhat more probability to more extreme
test-statistic values than the normal distribution would, reflecting
the honest fact that there's more that could go "wrong" (purely by
chance) when both the mean AND the spread have to be estimated from
the same limited data.

WHY A SMALL-SAMPLE CRITICAL VALUE IS LARGER THAN 1.96
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Because the t-distribution's tails are heavier (more spread out) than
the normal distribution's, a larger cutoff value is needed to still
capture the same 95% of the distribution's own probability in the
middle, leaving only 5% split between the two tails. At df=9, per this
chapter's own critical value, that cutoff is 2.262 - noticeably larger
than the z-test's fixed 1.96, because a small sample of only 10 data
points (n = df+1 = 10) gives a genuinely unreliable estimate of s, and
the wider critical value compensates for that unreliability by
requiring stronger evidence before rejecting H0.

WHY THE GAP SHRINKS AS SAMPLE SIZE GROWS
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Per this chapter's own convergence finding, as df (and therefore
sample size n) grows, the estimate s becomes a much more reliable
stand-in for the true sigma - there's simply more data to pin it down
accurately. With less extra uncertainty to account for, the t-
distribution's tails thin out and its shape moves closer and closer to
the normal distribution, which is exactly why this chapter's own
critical-value table shows the numbers steadily shrinking toward 1.96
as df increases (2.262 at df=9, down to 2.042 at df=30, and
essentially 1.96 for very large df).

WHY THIS WORKS AS AN ANSWER
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The explanation is grounded directly in this chapter's own stated
reason for the t-distribution's existence (the extra uncertainty from
estimating sigma via s), connects that reasoning concretely to why a
wider critical value is needed at small df, and explains the
convergence toward 1.96 using this chapter's own stated relationship
between growing sample size and a more reliable estimate of s.