91Ó°ÊÓ

Given data:

$z<2.85$

To determine each proportion corresponding to a given z value, we will use the Standard Normal table. The notation $P\left(\right)$indicates that we want to compute a probability or a proportion.

$\mathrm{P}(z<2.85)$. Here's a sketch of the intended proportion, with the shaded area of interest:

We find the z-value $2.8$on the left side of the Standard Normal table, then proceed to the right in that row until we reach the column headed by $05$. $0.9978$.

Given data:

$z>-1.66$.

This is the "opposite" of the percentage we discovered in part one (A). We shade the typical normal curve on the other side as follows:

Given data:

$z>-1.66$

$P(z>-1.66)$Here is the sketch of the desired proportion:

From the Standard Normal table, we look up the z-value $-1.6$on the left side of the table, and then move to the right in that row until we are under the column headed by $.06$. The proportion is read as $.0485$. But this is the proportion to the left of $-1.66$. To calculate the proportion to theright of $-1.66$, we simply subtract from $1$ as shown here:

Area to right $1-0.485$

$=.9515$

Given data:

$-1.66<z<2.85$

$P(-1.66<z<2.85)$.This is a "between" problem that will take several steps to solve. First, we'll make the sketch:

We now look at the sketch again, and we identify three different pieces of it: the unshaded piece to the left of $-1.66$, the shaded piece between $-1.66$and $2.85$, and the unshaded piece to the right of $2.85$.

We designate them as areas A. B, and C, respectively:

In part (C) we determined that area A was $.0485$, and in part (B) we found that area C was $.0022$.

We know that the total area, or proportion, under the curve is 1 . Thus, we know that $A+B+C=1$. We know A and C already. So it is a matter of subtraction to determine the value of area B, which is the area (proportion) of interest.

$A+B+C=1$

$.0485+B+.0022=1$

$B+.0507=1$

$B=.9493$

Area (proportion) B is $.9493$, which is the $P(-1.66<z<2.85)$.