Chapter 4.3, Problem 58E

There is no nice formula for the standard normal cdf Φ(z), but several good approximations have been published in articles. The following is from “Approximations for Hand Calculators

Using Small Integer Coefficients” (Mathematics of Computation, 1977: 214–222). For 0 < z ≤ 5.5,

     $\begin{array}{lll}P(Z\ge z)\hfill & =\hfill & 1-\Phi (z)\hfill \\ \hfill & \approx \hfill & .5\exp \left\{-\left[\frac{(83z+351)z+562}{703/z+165}\right]\right\}\hfill \end{array}$

The relative error of this approximation is less than .042%. Use this to calculate approximations to the following probabilities, and compare whenever possible to the probabilities obtained from Appendix Table A.3.

a. P(Z ≥ 1)

b. P(Z < −3)

c. P(−4 < Z < 4)

d. P(Z > 5)