Question

5. Determine the z-score value in each of the following scenarios:
a. What z-score value separates the top 8% of a normal distribution from the bottom
92%?
b. What z-score value separates the top 72% of a normal distribution from the bottom
28%?
c. What z-score value form the boundaries for the middle 58% of a normal
distribution?
d. What z-score value separates the middle 45% from the rest of the distribution?

Answer 1

Consider normal distribution:

$$p(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\bigg(-\frac{x^2}{2\sigma^2}\bigg)$$

and

$$F(x) = \int_{-\infty}^{x} p(t) \, dt$$

and the probability to find x in (a,b) can be found as

$$P(a<x<b) = F(b)-F(a)$$

and

$$P(x<b) = F(b)$$

due to

$$F(-\infty) = 0$$

Then we need to find the solutions for

a) P(x<z) = F(z) = 92%

b) P(x<z) = F(z) = 28%

c) P(-z<x<z) = F(z) - F(-z) = 58%

d) P(-z<x<z) = F(z) - F(-z) = 45%

Here

$$z = X \sigma$$

where X is found numerically (in Mathematica) to be equal

a) X = 1.406

b) X = -0.583

c) X = 0.806

d) X = 0.598