Q. 6.92 - Elementary Statistics

Question

A variable is normally distributed with a mean of $0$ and standard deviation $4.$ Find the percentage of all possible values of the variable.

a. lie between $-8$ and $8$ .

b. exceed $-1.5$,

c. are less than $2.75.$

Step-by-Step Solution

Answer

a) $95.44\%$

b) $64.80\%$

c) $74.59\%$

Step 1: Given Information (Part a)

To find the percentage of observations that lie between $-8$ and  $8.$

Step 2: Explanation (Part a)

Calculate the$z-$scores as follows:

 $$\begin{gathered} z=\frac{-8-0}{4} \\ =-2 \end{gathered}$$

$$\begin{gathered} z=\frac{8-0}{4} \\ =2 \end{gathered}$$

As a result, the observations between $-8$ and $-8$have the same$z-$scores as the$z-$ scores between $-2$ and 2.

The proportions that are less than the $z$- scores are taken from Table II in Appendix A.

$-2$and $-2$ are equal to$0.0228$and $0.9772$, respectively.

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

$0.9772-0.0228 = 0.9544$

$95.44$ percent of the time

Step 1: Given Information (Part b)

To find the percentage of observations that are greater than -1.5.

Step 2: Explanation(Part b)

Calculate the $z$ - score:

$$\begin{gathered} z=\frac{-1.5-0}{4} \\ \approx -0.38 \end{gathered}$$

As a result, observations greater than 5 correspond to $z$- scores greater than $-0.38.$

The proportion of $z-$ scores greater than $-0.38$ is shown in Table II of Appendix A.

$1-0.3520 = 0.6480$

$=64.80$

Step 1: Given Information (Part c)

To find The percentage of observations that are less than $2.75.$

Step 2: Explanation (Part c)

Determine the $z$ - score:

$$\begin{gathered} z=\frac{2.75-0}{4} \\ \approx 0.69 \end{gathered}$$

Therefore the observations less than $2.75$are the same as the $z$- scores less than $0.69$. From Table II in Appendix A, the proportion of $z$- scores less than $0.69$ is

$0.7459=74.59 \%.$