Question

4. For a normal distribution, find the z-score that separates the distribution as follows:

Answer 1

Let $\Phi(z)$ denote the cumulative distribution function for the standard normal distribution (mean 0, standard deviation 1). Then you're looking for z such that

(a) $\Phi(z) = P(Z\le z) = 0.70$

(b) $\Phi(z) = P(Z\le z) = 0.40$

(c) $\Phi(z) = P(Z\le z) = 0.25$

To get the corresponding z-score for each of 0.70, 0.40, and 0.25, take the inverse CDF of both sides:

(a) $z = \Phi^{-1}(0.70) \approx 0.5244$

(b) $z = \Phi^{-1}(0.40) \approx -0.2533$

(c) $z = \Phi^{-1}(0.25) \approx -0.6745$