Let X~N(?,?2), and let Z=(X??)/?. Use Equation (4.25) to

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

ISBN: 9780073401331 38

Solution for problem 19E Chapter 4.5

Statistics for Engineers and Scientists | 4th Edition

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Problem 19E

Let X ∼ N$\left(\mu, \sigma^{2}\right)$, and let Z = $(X-\mu) / \sigma$ . Use Equation (4.25) to show that Z ∼ N(0, 1).

Equation transcription:

$(\mu{},{\sigma{}}^{2})$

$(X-\mu{})/\sigma{}$

Text transcription:

(\mu, sigma^{2})

(X-\mu) / sigma

Step-by-Step Solution:

Answer:

Step 1 of 1:

In this question, we are asked to prove $Z\sim{}N(0,1)$ , using equation $4.25$ .

Let $X\ \sim{}N(\mu{},\ {\sigma{}}^{2})$ , and let $Z\ =\ \frac{X\ -\ \mu{}}{\sigma{}}$

Equation $4.25$ state that,

Let $X\ \sim{}N(\mu{},\ {\sigma{}}^{2})$ , and let $a\ \ne{}0$ and $b$ be constants. Then,

$aX+b\ \sim{}N(a\mu{}+b,{\ a}^{2}{\sigma{}}^{2})$ ………..(1)

We know that,

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

$Z\ =\ \frac{X}{\sigma{}}\ -\frac{\mu{}}{\sigma{}}$ ………(2)

From equation (2), let,

$a\ =\frac{1}{\sigma{}},\ and$ $b\ =\ -\frac{\mu{}}{\sigma{}}\$

Now we can substitute value of $a$ and $b$ ,

$a\mu{}+b\ =$ $\frac{\mu{}}{\sigma{}}+(-\frac{\mu{}}{\sigma{}})\ \ =0$

${a}^{2}{\sigma{}}^{2}\ ={\left({\frac{1}{\sigma{}}}\right)}^{2}\times{}{\sigma{}}^{2}\ =1\$

Substitute the value of $a$ , $b$ , $a\mu{}+b$ and ${a}^{2}{\sigma{}}^{2}$ into equation (1),

$aX+b\ \sim{}N(a\mu{}+b,{\ a}^{2}{\sigma{}}^{2})$

$\frac{X}{\sigma{}}+(-\frac{\mu{}}{\sigma{}})\sim{}N(0\ ,\ 1)$

$\ \frac{X\ -\ \mu{}}{\sigma{}}\ \sim{}\ N(0\ ,\ 1)$

$\ Z\ \sim{}\ N(0\ ,1)$

Hence proved.

Step 2 of 1

Chapter 4.5, Problem 19E is Solved

Textbook: Statistics for Engineers and Scientists

Edition: 4

Author: William Navidi

ISBN: 9780073401331

Since the solution to 19E from 4.5 chapter was answered, more than 426 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 19E from chapter: 4.5 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM. The answer to “?Let X ? N$\left(\mu, \sigma^{2}\right)$, and let Z = $(X-\mu) / \sigma$ . Use Equation (4.25) to show that Z ? N(0, 1).Equation transcription:Text transcription:(\mu, sigma^{2})(X-\mu) / sigma” is broken down into a number of easy to follow steps, and 28 words. Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. This full solution covers the following key subjects: let, show, equation, use. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4.