Question

percentages to z - scores

1. find the value z of a standard normal distribution that satisfy each of the following conditions.

(a) 28% of the observations fall below z.
(b) 60% of the observations fall above z.
(c) the middle 70% of observations correspond to what two z - scores?

Explanation:

Step1: Use the standard normal distribution table (a)

We know that \(P(Z < z)=0.28\). Looking up the value \(0.28\) in the standard - normal distribution table (the body of the table), the closest value to \(0.28\) is \(0.2810\) which corresponds to \(z=- 0.58\).

Step2: Use the property of the standard normal distribution (b)

If \(P(Z>z) = 0.6\), then \(P(Z\leq z)=1 - 0.6=0.4\). Looking up the value \(0.4\) in the standard - normal distribution table, the closest value to \(0.4\) is \(0.4013\) which corresponds to \(z=-0.25\)

Step3: Use the property of the middle - area of the standard normal distribution (c)

If the middle area is \(0.7\), then the area in the two tails is \(1 - 0.7 = 0.3\). The area in each tail is \(\frac{0.3}{2}=0.15\).

We want to find \(z_1\) and \(z_2\) such that \(P(Z < z_1)=0.15\) and \(P(Z < z_2)=1 - 0.15 = 0.85\)

Looking up \(0.15\) in the standard - normal distribution table, the closest value to \(0.15\) is \(0.1492\) which corresponds to \(z_1=-1.04\)

Looking up \(0.85\) in the standard - normal distribution table, the closest value to \(0.85\) is \(0.8508\) which corresponds to \(z_2 = 1.04\)

Answer:

(a) \(z=-0.58\)

(b) \(z=-0.25\)

(c) \(z=-1.04\) and \(z = 1.04\)