Algebra and Trigonometry (6th Edition)
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Author: Robert F. Blitzer
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![### Z-Score and Normal Distribution Calculation
**Question:**
A normal distribution has a data point 28.2 with a z-score of 1.4. If the standard deviation is 3.8, what is the mean?
**Options:**
- 22.88
- 28.2
- 33.52
- 11.75
**Detailed Explanation:**
Let’s use the z-score formula to calculate the mean, where:
\[ \text{z-score} = \frac{(x - \mu)}{\sigma} \]
Given:
- Data point \( x = 28.2 \)
- Z-score \( z = 1.4 \)
- Standard deviation \( \sigma = 3.8 \)
We need to find the mean \( \mu \). Rearrange the formula to solve for \( \mu \):
\[ \mu = x - z \cdot \sigma \]
Substitute the given values into the equation:
\[ \mu = 28.2 - 1.4 \cdot 3.8 \]
\[ \mu = 28.2 - 5.32 \]
\[ \mu = 22.88 \]
Therefore, the mean of the distribution is **22.88**.
**Correct Answer: 22.88**](https://content.bartleby.com/qna-images/question/253ec4e6-fefa-47c2-a875-f6d418ff65f8/ae4f0d64-1bff-4fe7-ba9e-9c9e17213dfa/rn18ktm_processed.jpeg)
Transcribed Image Text:### Z-Score and Normal Distribution Calculation
**Question:**
A normal distribution has a data point 28.2 with a z-score of 1.4. If the standard deviation is 3.8, what is the mean?
**Options:**
- 22.88
- 28.2
- 33.52
- 11.75
**Detailed Explanation:**
Let’s use the z-score formula to calculate the mean, where:
\[ \text{z-score} = \frac{(x - \mu)}{\sigma} \]
Given:
- Data point \( x = 28.2 \)
- Z-score \( z = 1.4 \)
- Standard deviation \( \sigma = 3.8 \)
We need to find the mean \( \mu \). Rearrange the formula to solve for \( \mu \):
\[ \mu = x - z \cdot \sigma \]
Substitute the given values into the equation:
\[ \mu = 28.2 - 1.4 \cdot 3.8 \]
\[ \mu = 28.2 - 5.32 \]
\[ \mu = 22.88 \]
Therefore, the mean of the distribution is **22.88**.
**Correct Answer: 22.88**
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