Calculating Proportions: The TI84 (and other models) has a...

Calculating Proportions: The TI84 (and other models) has a built-in function Normalcdf that calculates proportions based on normal distributions. Z Distribution 1. P(0 < z < 1) = 4. P(z > 1) = 2. P(-1 < z < 2) = 5. P(z < 0.5) = 3. P(1.2 < z < 2.4) = 6. Find the 90$^{th}$ percentile for z-scores. P(a < z < b) = Normalcdf(a, b)

There is no nice formula for the standard normal cdf $\Phi(z),$ but several good approximations have been published in articles. The following is from "Approximations for Hand Calculators Using Small Integer Coefficients" (Math. Comput, $1977 : 214-222 ) .$ For $0 < z \leq 5.5$ , $$P(Z \geq z)=1-\Phi(z) \approx .5 \exp \left\{-\left[\frac{(83 z+351) z+562}{(703 / z)+165}\right]\right\}$$ 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) $P(Z \geq 1)$ (b) $P(Z<-3)$ (c) $P(-4< Z<4)$ (d) $P(Z > 5)$

Probability with Applications in Engineering, Science, and Technology

Continuous Random Variables and Probability Distributions

The Normal (Gaussian) Distribution

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