Exact answer: 6.038 minutes.
prior precision = 1 / 1.0 = 1
data precision = n / σ² = 9 / 4 = 2.25
posterior precision = 1 + 2.25 = 3.25
posterior mean = (5.0 × 1 + 6.5 × 2.25) / 3.25
= (5.0 + 14.625) / 3.25
= 19.625 / 3.25
≈ 6.038
If your guess landed noticeably closer to 6.5 than to 5.0, your intuition for precision-weighting is
already on track — the data’s precision (2.25) is more than double the prior’s (1), so it should pull
harder. The posterior mean sits about 69% of the way from the prior mean to the sample mean
((6.038 − 5.0) / (6.5 − 5.0) = 1.038 / 1.5 ≈ 0.69) — proportional to the data’s share of total
precision (2.25 / 3.25 ≈ 0.69, exactly).
Why this is conjugacy, concretely. Just like beta-binomial, the update didn’t require touching
an integral: you added two precisions and took a weighted average of two means, and the result was
guaranteed (by the algebra of normal distributions) to itself be normal. That’s what “conjugate”
buys you — a prior family paired with a likelihood family such that the posterior stays in the same
family, so the update reduces to arithmetic on the family’s parameters. Beta-binomial adds counts
(α, β); normal-normal adds precisions and takes a weighted mean. Different families, same
underlying shape: combine what you believed with what you saw, weighted by how much each one is
worth trusting.
More data narrows things further, exactly like Beta did. If this had been n = 900 sessions
instead of 9 (same sample mean), data precision would be 900/4 = 225, dwarfing the prior’s 1 — the
posterior mean would land almost exactly on 6.5, and the posterior variance (1/posterior precision)
would shrink correspondingly. Same “sharpening” phenomenon from Lesson 16, different distribution
family.
Where this goes: beta-binomial and normal-normal are two members of a broader family of conjugate pairs (gamma-Poisson is another common one you’ll meet if you go looking). But conjugate pairs are the exception, not the rule — most real priors and likelihoods you’ll want to combine don’t have a tidy closed form. Next lesson previews what you do when the closed form runs out.