Normal distribution

z = (x − μ)/σ turns any normal-distribution question into a standard one. Appears every year, and is easy to mistake for a topic outside "probability".

The two directions

Forward (find a probability) z = (x − μ)/σ, then read P(Z < z) or P(Z > z) Sketch the curve and shade the region asked for before calculating Inverse (find a value or σ) Given a probability, find z first (inverse normal), then solve x = μ + zσ for the unknown A LEFT-tail probability gives a NEGATIVE z Exam-style μ = 118 g, σ = 19 g; 10% of muffins are below the minimum weight Find the maximum weight of a muffin that cannot be sold. P(X < x) = 0.10 is a left-tail probability, so z is negative: inverse normal gives z ≈ −1.281 z = (x − μ)/σ → −1.281 = (x − 118)/19 x = 118 + (−1.281 × 19) ≈ 93.7 g ✓ ≈ 93.7 g — muffins below this weight are the bottom 10%. ✕

"Not appropriate" needs a context reason, not just a claim

When asked why a normal distribution might NOT be a good model, a bare "it isn't symmetrical" earns little. Tie the reason to the CONTEXT: a large standard deviation relative to the mean implies the left tail goes below a value that is impossible in real life (negative hours, negative minutes) — or a cluster of people at exactly zero makes the real shape skewed rather than bell-shaped and symmetrical.

The two most common context-based reasons

A large σ relative to μ implies part of the curve extends below a real-world impossible value (negative time, negative weight).
A cluster of the population at exactly zero (or another natural boundary) makes the true shape skewed rather than bell-shaped.

Practice for this standard covers the whole topic — flashcards and quiz.