Question

Find the specified areas for a \(N(0,1)\) density. (a) The area below \(z=0.8\) (b) The area above \(z=1.2\) (c) The area between \(z=-1.75\) and \(z=-1.25\)

Short answer

The area below \(z=0.8\) is 0.7881, the area above \(z=1.2\) is 0.1151, and the area between \(z=-1.75\) and \(z=-1.25\) is 0.0655.

Step-by-step solution

Interpret the tasks

To solve this exercise, each task should first be interpreted in light of the normal distribution curve: (a) The area below \(z=0.8\) refers to all values less than 0.8 under the curve. (b) The area above \(z=1.2\) refers to all values greater than 1.2. (c) The area between \(z=-1.75\) and \(z=-1.25\) refers to all values that lie within these two z-scores.

Find the areas using a Z-table or calculator

(a) From the Standard Normal (Z-) Table, the area below \(z=0.8\) can be found directly: it is 0.7881. (b) The area above \(z = 1.2\) is found by subtracting the cumulative area under the curve up to \(z = 1.2\) from 1. From the Z-Table, the cumulative area up to \(z = 1.2\) is 0.8849. Thus, the area above \(z = 1.2\) is \(1 - 0.8849 = 0.1151\). (c) The area between \(z=-1.75\) and \(z=-1.25\) is found by subtracting the cumulative area up to \(z = -1.75\) from the cumulative area up to \(z = -1.25\). From the Z-Table, these areas are 0.0401 and 0.1056, respectively. Thus, the area between \(z=-1.75\) and \(z=-1.25\) is \(0.1056 - 0.0401 = 0.0655\).