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.
