The standard deviation of a Normal distribution can be estimated with the “Sample standard deviation, “s”.

s is a biased = incorrect estimator of σ, especially biased at small n values.

Averages of 400,000 estimates of s, from a Normal distribution with σ = 1.

t testing was developed because Z testing was very inaccurate at small n values. The sole cause of this inaccuracy is the inaccuracy of s at small n values. The t distributions correct s to σ;

σ = calculated s / table s. P (t ≤ t test) = P (Z ≤ Z test) for any set of x̄ 1, x̄ 2, σ, s, and n.

Z test = (x̄ 1 - x̄ 2) / (σ / √ n)

With Z tests, “σ must be known”. This means that at small n values, using s in the formula, Z is inaccurate; Z is only correct when σ or s corrected to σ is used, or when n is large enough that s ≈ σ and s is used-although Z test is then (slightly) incorrect.

“The Z test is for n > 30.” σ is known, thus the Z test is perfectly accurate at any n.

t test = (x̄ 1 - x̄ 2) / (s / √ n)

“With t tests, σ is unknown”. This means that s must be used in the t test calculations; use of another estimator of σ, or of σ, makes t test incorrect.

“The t test is for n < 30”. The t test is as perfectly accurate at any n.
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