We appreciate your visit to Assume the distribution is normal or approximately normal and calculate the percentage using the tex 68 95 99 7 tex rule Given a mean of. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!
Answer :
To solve this problem, we use the properties of the normal distribution and the empirical rule, also known as the 68-95-99.7 rule. This rule states the following for a normal distribution:
- About 68% of the data falls within 1 standard deviation from the mean.
- About 95% of the data falls within 2 standard deviations from the mean.
- About 99.7% of the data falls within 3 standard deviations from the mean.
Given:
- Mean ([tex]\(\mu\)[/tex]) = 23
- Standard Deviation ([tex]\(\sigma\)[/tex]) = 157
- We need to find the percentage of the distribution that lies between 23 and 337.
1. Calculate the Z-scores:
- The Z-score measures how many standard deviations a data point is from the mean.
- For the lower bound (23):
[tex]\[
Z = \frac{\text{lower bound} - \text{mean}}{\text{standard deviation}} = \frac{23 - 23}{157} = 0
\][/tex]
- For the upper bound (337):
[tex]\[
Z = \frac{\text{upper bound} - \text{mean}}{\text{standard deviation}} = \frac{337 - 23}{157} = 2
\][/tex]
2. Interpreting the Z-scores:
- The Z-score for the lower bound is 0, which means it is exactly at the mean.
- The Z-score for the upper bound is 2, which means it is 2 standard deviations above the mean.
3. Applying the 68-95-99.7 Rule:
- Since the distance from the mean to 337 is 2 standard deviations (as indicated by the Z-score of 2), the empirical rule tells us that approximately 95% of the data in a normal distribution falls within 2 standard deviations of the mean.
Therefore, approximately 95% of the distribution lies between 23 and 337.
- About 68% of the data falls within 1 standard deviation from the mean.
- About 95% of the data falls within 2 standard deviations from the mean.
- About 99.7% of the data falls within 3 standard deviations from the mean.
Given:
- Mean ([tex]\(\mu\)[/tex]) = 23
- Standard Deviation ([tex]\(\sigma\)[/tex]) = 157
- We need to find the percentage of the distribution that lies between 23 and 337.
1. Calculate the Z-scores:
- The Z-score measures how many standard deviations a data point is from the mean.
- For the lower bound (23):
[tex]\[
Z = \frac{\text{lower bound} - \text{mean}}{\text{standard deviation}} = \frac{23 - 23}{157} = 0
\][/tex]
- For the upper bound (337):
[tex]\[
Z = \frac{\text{upper bound} - \text{mean}}{\text{standard deviation}} = \frac{337 - 23}{157} = 2
\][/tex]
2. Interpreting the Z-scores:
- The Z-score for the lower bound is 0, which means it is exactly at the mean.
- The Z-score for the upper bound is 2, which means it is 2 standard deviations above the mean.
3. Applying the 68-95-99.7 Rule:
- Since the distance from the mean to 337 is 2 standard deviations (as indicated by the Z-score of 2), the empirical rule tells us that approximately 95% of the data in a normal distribution falls within 2 standard deviations of the mean.
Therefore, approximately 95% of the distribution lies between 23 and 337.
Thanks for taking the time to read Assume the distribution is normal or approximately normal and calculate the percentage using the tex 68 95 99 7 tex rule Given a mean of. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
- Why do Businesses Exist Why does Starbucks Exist What Service does Starbucks Provide Really what is their product.
- The pattern of numbers below is an arithmetic sequence tex 14 24 34 44 54 ldots tex Which statement describes the recursive function used to..
- Morgan felt the need to streamline Edison Electric What changes did Morgan make.
Rewritten by : Barada
