Q. 6.7 - Elementary Statistics

Question

The total area under the following standard normal curve is divided into eight regions.

a. Determine the area of each region.

b. Complete the following table.

Step-by-Step Solution

Answer

(a) The area of each section will be

       Area at $1$ standard deviation $=34.13\%$

       Area at $2$ standard deviation $=13.59\%$

       Area at $3$ standard deviation $=2.28\%$

(b) The area of each section will be

        $\begin{array}{ccc}\text{ Region }& \text{ Area }& \begin{array}{c}\text{ Persentage of }\\ \text{ total area }\end{array}\\ -\infty \text{ to }-3& 0.0013& 0.13\\ -3\text{ to }-2& 0.0215& 2.15\\ -2\text{ to }-1& 0.1359& 13.59\\ -1\text{ to }0& 0.3413& 34.13\\ 0\text{ to }1& 0.3413& 34.13\\ 1\text{ to }2& 0.1359& 13.59\\ 2\text{ to }3& 0.0215& 2.15\\ 3\text{ to }\infty & 0.0013& 0.13\\ & 1.0000& 100.00\end{array}$

Part(a) Step 1: Given Information

Part(a) Step 2: Explanation

Using the relation, calculate the standard deviation and mean.

$\sigma =\sqrt{\frac{\sum {\left(x_i-\overline{x}\right)}^2}{n-1}}$

$\overline{x}=\frac{\sum x_i}{n}$

We'll get it after we've solved it.

$\sigma =1.78$

$\overline{x}=0$

Then, using the illustration as a guide, determine the area.

Part(b) Step 1: Given Information

Part(b) Step 2: Explanation

Using the relation, calculate the standard deviation and mean.

$\sigma =\sqrt{\frac{\sum {\left(x_i-\overline{x}\right)}^2}{n-1}}$

$\overline{x}=\frac{\sum x_i}{n}$

We'll get it after we've solved it.

$\sigma =1.78$

$\overline{x}=0$

Then, using the illustration as a guide, determine the area.