Q. 6.89 - Elementary Statistics

Question

A variable is normally distributed with a 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 $4$.

Step-by-Step Solution

Answer

(a) The percentage of all possible values of the variable that lie between $1$ and $7$ is $68.53\%.$

(b) The percentage of all possible values of the variable that exceed $5$ is $69.15\%.$

(c) The percentage of all possible values of the variable that are less than $4$is$15.87\%.$

Part (a) Step 1: Given information

The given mean is $6$

Standard deviation is $2.$

Part (a) Step 2: Explanation

The given data is written as 

Mean, $\mu =6$

Standard deviation, $\sigma =2$

The random variable $X$ has a $z-$score which is given as

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

Find the $z-$scores

For 1, 

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

    $=-2.5$

For $7$, 

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

    $=0.5$

Therefore the observations between $1$ and $7$ are the same as the $z-$ scores between $-2.5$ and $0.5$.

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

$0.6915-0.0062=0.6853$

                               

Part (b) Step 1: Given information

The given mean is $6$

Standard deviation is $2.$

Part (b) Step 2: Explanation

Use the above formula and find the $z-$score

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

   $=-0.5$

As a result, observations more than $5$ correspond to z scores greater than $-0.5$.

The proportion of $z-$scores greater than $-0.5$ is calculated in Table II of Appendix A

$1-0.3085=0.6915$

                    $=69.15\%.$

Part (c) Step 1: Given information

The given mean is $6$

Standard deviation is $2.$

Part (c) Step 2: Explanation

The $z-$score is obtained by

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

    $=-1.$

Therefore the observations less than $4$ are the same $z-$scores less than $-1$. From Table II in Appendix A, the proportion of $z-$ scores less than $-1$ is

$0.1587=15.87\%.$