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 =
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.3: Measures Of Spread
Problem 26PFA
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![### 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 =
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