t-Distribution vs z

Drag degrees of freedom (df = n − 1). The teal t curve grows taller and narrower toward the dashed navy standard Normal z. Because t has heavier tails, its critical value t* exceeds z* — and shrinks toward z* as df grows. This is why means inference uses t.

Student t (df)    standard Normal z    tail area under t

critical t* (df = 5)
critical z*
t* − z* (extra width)

Try: set df = 1 — the two-tail t* is about 12.7, far above z* = 1.96, with hugely heavy tails. Slide df up: by df = 30 the curves nearly coincide and t* − z* drops under 0.1. The AP exam always uses t for means inference regardless.