Chapter 2, Problem 7CRE

(a)

To determine

To find: the proportion of observations from a standard Normal distribution

Answer to Problem 7CRE

Hence, there exists $12.2\%$ of observations are in the region $z\le -2.25$ .

Explanation of Solution

Given:

  $z\le -2.25$

Calculation:

The given region is shown in the following graph as follows:

  

From standard normal table, the value below -2.25 is 0.0122

Hence, there exists $12.2\%$ of observations are in the region $z\le -2.25$ .

Conclusion:

Hence, there exists $12.2\%$ of observations are in the region $z\le -2.25$ .

(b)

To determine

To find: the proportion of observations from a standard Normal distribution

Answer to Problem 7CRE

The proportion of observation is $0.9878$

Explanation of Solution

Given:

  $z\ge -2.25$

Calculation:

Find the proportion of observations from Standard Normal distribution for the region

  $z\ge -2.25$

The given region is shown in the following graph as follows.

  

From part (a), the area below the value -2.25 is 0.0122

Then, the proportion of observations larger than -2.25 is calculated as follows.

=Total Proportion- proportion of observations before -2.25

  $\begin{array}{l}=1-0.0122\hfill \\ =0.9878\hfill \end{array}$

Conclusion:

The proportion of observation is $0.9878$

(c)

To determine

To find: the proportion of observations from a standard Normal distribution

Answer to Problem 7CRE

Hence, $3.84\%$ of observations are above 1.77

Explanation of Solution

Given:

  $z>1.77$

Calculation:

Find the proportion of observations from Standard Normal distribution for the region

  $z>1.77$

The given region is shown in the following graph as follows.

  

Using Standard Normal tables, the area before 1.77 is 0.9616. Therefore, the proportion of observations below 1.77 is 0.9616

Then, the proportion of observations greater than 1.77 is the value of proportion less than 1.77subtracted from 1 (total of the proportion)

That is, the proportion of observations greater than 1.77

  $1-0.9616=0.0384=3.84\%$

Hence, $3.84\%$ of observations are above 1.77

Conclusion:

Hence, $3.84\%$ of observations are above 1.77

(d)

To determine

To find: the proportion of observations from a standard Normal distribution

Answer to Problem 7CRE

Hence, $94.94\%$ of observations are between -2.25 and 1.77

Explanation of Solution

Given:

  $-2.25<z<1.77$

Calculation:

Find the proportion of observations from Standard Normal distribution for the region

  $-2.25<z<1.77$

The given region is shown in the following graph as follows:

  

The required region can be obtained by subtracting the area before -225 from the area before

1.77

From part (a), the area before -2.25 is 0.0122. Also, the area before 1.77 is 0.9616 (from

Standard Normal tables)

Then, the area between -2.25 and 1.77 is

  $0.9616-0.0122=0.9494=94.94\%$

Hence, $94.94\%$ of observations are between -2.25 and 1.77

Conclusion:

Hence, $94.94\%$ of observations are between -2.25 and 1.77