A sample consists of the following n = 6 scores: 5, 5, 5, 8, 8, 8 Transform the distribution so that it has a new mean of M = 100 and standard deviation of s = 20. What is the new, transformed score for the original score of X = 5? new X =
A sample consists of the following n = 6 scores: 5, 5, 5, 8, 8, 8 Transform the distribution so that it has a new mean of M = 100 and standard deviation of s = 20. What is the new, transformed score for the original score of X = 5? new X =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![### Statistical Transformation Example
#### Sample Data:
A sample consists of the following \( n = 6 \) scores:
\[ 5, 5, 5, 8, 8, 8 \]
#### Task:
Transform the distribution so that it has a new mean of \( M = 100 \) and a standard deviation of \( s = 20 \).
#### Question:
What is the new, transformed score for the original score of \( X = 5 \)?
**New \( X \) = [__________]**
---
### How to Transform the Scores
To transform the scores to a new distribution with a specified mean (M) and standard deviation (s), you can use the following steps:
1. Calculate the original mean (\( \bar{X} \)) and standard deviation (\( s_X \)) of the sample.
2. Use the formula for transforming each score:
\[ \text{New } X = M + \left( \frac{X - \bar{X}}{s_X} \right) \times s \]
Where:
- \( X \) is the original score.
- \( \bar{X} \) is the original mean.
- \( s_X \) is the original standard deviation.
- \( M \) is the new mean.
- \( s \) is the new standard deviation.
### Example Calculation:
1. **Calculate the original mean (\( \bar{X} \)):**
\[ \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n} = \frac{5 + 5 + 5 + 8 + 8 + 8}{6} = 6.5 \]
2. **Calculate the original standard deviation (\( s_X \)):**
\[ s_X = \sqrt{\frac{\sum (X_i - \bar{X})^2}{n}} = \sqrt{\frac{(5 - 6.5)^2 + (5 - 6.5)^2 + (5 - 6.5)^2 + (8 - 6.5)^2 + (8 - 6.5)^2 + (8 - 6.5)^2 }{6}} = 1.5 \]
3. **Transform the original score \( X = 5 \) using the new parameters \( M = 100 \) and \( s =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2a79f3c-9d5f-475d-8512-37c872933c1c%2F96e47ed0-e9bd-4f25-9c9d-d2273a3bf01b%2F0a2g3q_processed.png&w=3840&q=75)
Transcribed Image Text:### Statistical Transformation Example
#### Sample Data:
A sample consists of the following \( n = 6 \) scores:
\[ 5, 5, 5, 8, 8, 8 \]
#### Task:
Transform the distribution so that it has a new mean of \( M = 100 \) and a standard deviation of \( s = 20 \).
#### Question:
What is the new, transformed score for the original score of \( X = 5 \)?
**New \( X \) = [__________]**
---
### How to Transform the Scores
To transform the scores to a new distribution with a specified mean (M) and standard deviation (s), you can use the following steps:
1. Calculate the original mean (\( \bar{X} \)) and standard deviation (\( s_X \)) of the sample.
2. Use the formula for transforming each score:
\[ \text{New } X = M + \left( \frac{X - \bar{X}}{s_X} \right) \times s \]
Where:
- \( X \) is the original score.
- \( \bar{X} \) is the original mean.
- \( s_X \) is the original standard deviation.
- \( M \) is the new mean.
- \( s \) is the new standard deviation.
### Example Calculation:
1. **Calculate the original mean (\( \bar{X} \)):**
\[ \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n} = \frac{5 + 5 + 5 + 8 + 8 + 8}{6} = 6.5 \]
2. **Calculate the original standard deviation (\( s_X \)):**
\[ s_X = \sqrt{\frac{\sum (X_i - \bar{X})^2}{n}} = \sqrt{\frac{(5 - 6.5)^2 + (5 - 6.5)^2 + (5 - 6.5)^2 + (8 - 6.5)^2 + (8 - 6.5)^2 + (8 - 6.5)^2 }{6}} = 1.5 \]
3. **Transform the original score \( X = 5 \) using the new parameters \( M = 100 \) and \( s =
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 1 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman