What is the percentage of values in the distribution between 7 and 13? %
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![### Understanding Normal Distribution and the 68-95-99.7 Rule
A normal distribution has a mean of 7 and a standard deviation of 2. Use the 68-95-99.7 rule to find the percentage of values in the distribution between 7 and 13.
---
**Question:**
What is the percentage of values in the distribution between 7 and 13?
[ ] %
---
#### Explanation:
To answer this question, you need to understand the 68-95-99.7 rule, which is also known as the empirical rule in statistics.
1. **68% Rule:** This states that approximately 68% of the data within a normal distribution lies within one standard deviation of the mean.
2. **95% Rule:** This states that approximately 95% of the data lies within two standard deviations of the mean.
3. **99.7% Rule:** This states that approximately 99.7% of the data lies within three standard deviations of the mean.
Given:
- Mean (μ) = 7
- Standard Deviation (σ) = 2
For the distribution between 7 and 13:
- 7 is the mean.
- 13 is 3 standard deviations above the mean (7 + 3*2 = 13).
According to the 68-95-99.7 rule:
- 50% of the data is below the mean (since the normal distribution is symmetrical).
- 68% of the data lies within one standard deviation from the mean (7 ± 2).
Therefore, the percentage of values between 7 and 13 would include half of the 68% and all values from the mean to 3 standard deviations above the mean.
To calculate this:
- 34% of values lie between 7 and 9 (mean to one standard deviation above mean).
- Values from 9 to 13 are two more standard deviations (86.6% - 34% already includes values from mean to one standard deviation above mean).
Adding these percentages together:
- 34% (mean to one standard deviation) + 47.7% (second and third standard deviation above mean) = 81.7%
Thus, the percentage of values in the distribution between 7 and 13 is approximately **81.7%**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F128593f5-7ea4-4e3b-94cd-55abf91c7165%2F67ec777a-3c60-4692-bebd-8a961b7a510d%2Fto2tfue_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Normal Distribution and the 68-95-99.7 Rule
A normal distribution has a mean of 7 and a standard deviation of 2. Use the 68-95-99.7 rule to find the percentage of values in the distribution between 7 and 13.
---
**Question:**
What is the percentage of values in the distribution between 7 and 13?
[ ] %
---
#### Explanation:
To answer this question, you need to understand the 68-95-99.7 rule, which is also known as the empirical rule in statistics.
1. **68% Rule:** This states that approximately 68% of the data within a normal distribution lies within one standard deviation of the mean.
2. **95% Rule:** This states that approximately 95% of the data lies within two standard deviations of the mean.
3. **99.7% Rule:** This states that approximately 99.7% of the data lies within three standard deviations of the mean.
Given:
- Mean (μ) = 7
- Standard Deviation (σ) = 2
For the distribution between 7 and 13:
- 7 is the mean.
- 13 is 3 standard deviations above the mean (7 + 3*2 = 13).
According to the 68-95-99.7 rule:
- 50% of the data is below the mean (since the normal distribution is symmetrical).
- 68% of the data lies within one standard deviation from the mean (7 ± 2).
Therefore, the percentage of values between 7 and 13 would include half of the 68% and all values from the mean to 3 standard deviations above the mean.
To calculate this:
- 34% of values lie between 7 and 9 (mean to one standard deviation above mean).
- Values from 9 to 13 are two more standard deviations (86.6% - 34% already includes values from mean to one standard deviation above mean).
Adding these percentages together:
- 34% (mean to one standard deviation) + 47.7% (second and third standard deviation above mean) = 81.7%
Thus, the percentage of values in the distribution between 7 and 13 is approximately **81.7%**.
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