6.89. A variable is normally distributed with mean $6$ and standard deviation $2$. Find the percentage of all possible values of the variable that
a. lie between $1$ and $7$.
b. exceed $5$ .
c. are less than $4$ .

To find the percentage of all possible values of the variable that lie between $1$ and $7$.

The $z$- scores is determined as follows:

$z=\frac{x-\mu }{\sigma }$

Here, mean is 6, and the standard deviation is 2.

For $1$is determine as follows:

$z=\frac{1-6}{2}$

$=-2.5$

For $7$is determine as follows:

$z=\frac{7-6}{2}$

$=0.5$

As a result, the $z$- scores between $-2.5$and $0.5$are the same as the observations between 1 and 7.

The proportions that are smaller than the $z-$scores $-2.5$and $0.5$are $0.0062$and $0.6915$, respectively, according to Table II in Appendix A.

The difference between the numbers in Table II is then used to calculate the percentage of all observations:

$0.6915-0.0062=0.6853$

$=68.53\%$

To find the percentage of all possible values of the variable that exceed $5$.

The $z$- score is determined as follows:
$z=\frac{x-\mu }{\sigma }$

$z=\frac{5-6}{2}$

$z=-0.5$

As a result, $z-$scores more than $-0.5$are the same for observations greater than $5$.

The fraction of $z-$scores greater than $-0.5$is shown in Table $II$in Appendix $A$.

$1-0.3085=0.6915$

$=69.15\%$

To find the percentage of all possible values of the variable that are less than $4$.

The $z-$ score is determined as follows:
$z=\frac{x-\mu }{\sigma }$

$z=\frac{4-6}{2}$$z=\frac{4-6}{2}$

$z=-1$

As a result, $z-$less than $-1$are the same as observations less than 4.

The proportion of $z-$scores smaller than $-1$is calculated in Table II in Appendix A.

$0.1587=15.87\%$