Stat question 4

Question

About what percentage of the area under the curve of the standard normal distribution is outside the interval z=[−2.96,2.96]z=[-2.96,2.96] (or beyond 2.96 standard deviations of the mean).

Answer 1

You would look up the cumulative probabilities Φ(z) = P(Z ≤ z) in the table. Note that since the normal distribution family is continuous, P(Z < z) = P(Z ≤ z). because P(Z = z) = 0 (point-events are null).

So inside the interval:

P(-2.96 ≤ z ≤ 2.96) = Φ(2.96) - Φ(-2.96) = 0.99846 - 0.00154 = 0.99692

This means outside we have 1 - 0.99692 = 0.00308

This is about 0.308%

Alternative method:

By symmetry, P(Z > 2.96) = P(Z < -2.96) = 0.00154

So P(|Z| > 2.96) = 2Φ(-2.96) = 2*0.00154 = 0.00308, which is again 0.308%.