Question

Find the specified areas for a \(N(0,1)\) density. (a) The area above \(z=-2.10\) (b) The area below \(z=-0.5\) (c) The area between \(z=-1.5\) and \(z=0.5\)

Short answer

So (a) The area above \(z=-2.10\) is 0.9821, (b) The area below \(z=-0.5\) is 0.3085, (c) The area between \(z=-1.5\) and \(z=0.5\) is 0.6247

Step-by-step solution

Finding the Area Above \(z=-2.10\)

The z-table shows the area to the left. Since the question asks for the area above \(z=-2.10\) which is actually the area to the right of \(z=-2.10\), we need to subtract the value in the z-table from 1 (all possible area under the curve). Therefore, the area above \(z=-2.10\) is \(1- z(-2.10)\). According to the z-table, \(z(-2.10)=0.0179\). Therefore the area above \(z=-2.10\) is \(1-0.0179=0.9821\)

Finding the Area Below \(z=-0.5\)

The z-table shows the area to the left so you can take the value straight from the table. Therefore, the area below \(z=-0.5\) is simply \(z(-0.5)\). According to the z-table, \(z(-0.5)=0.3085\). Thus, the area below \(z=-0.5\) is \(0.3085\)

Finding the Area Between \(z=-1.5\) and \(z=0.5\)

To find the area between \(z=-1.5\) and \(z=0.5\), we need to subtract the area to the left of \(z=-1.5\) from the area to the left of \(z=0.5\). So, the area between these two z-scores is \(z(0.5) - z(-1.5)\). According to the z-table, \(z(0.5)=0.6915\) and \(z(-1.5)=0.0668\). So, the area between \(z=-1.5\) and \(z=0.5\) is \(0.6915 - 0.0668=0.6247\)