Question

1. What is the area underneath every normal curve? 2. Find area between -2 and 2 in the standard normal distribution. 3. Describe how to transform a non-standard normal observation to a standard normal observation.

Answer 1

(1)

Area underneath every normal curve = 1

REASON: The area under the density curve is equal to 100% of all probabilities. So, the area underneath every normal curve = 1.

(2)

To find the area between - 2 and 2 in the standard normal distribution.

i.e.,

To find P(-2 < Z < 2):

Case 1: For Z from - 2 to mid value:
Table of Area Under Standard Normal Curve gives area = 0.4772

Case 2: For Z from mid value to 2:
Table of Area Under Standard Normal Curve gives area = 0.4772

Thus, area between - 2 and 2 is given by:

0.4772 X 2 = 0.9544

So

Answer is:

0.9544

(3)

To transform a non-standard normal observation X to a normal observation Z, the following Transformation is employed:

$Z=\frac{X-\mu }{\sigma }$

where

$\mu$ is the mean of the non-standard normal distribution and $\sigma$ is the standard deviation of the non-standard normal distribution.