Answer :
To solve this problem, we use the 68-95-99.7 rule, also known as the Empirical Rule, which helps us understand the distribution of data in a normal distribution.
Here's the step-by-step explanation:
Identify the Given Information:
- Mean ([tex]\mu[/tex]) = 24
- Standard Deviation ([tex]\sigma[/tex]) = 7
- We need to find the percentage of values above 17.
Determine How Many Standard Deviations 17 is From the Mean:
- We calculate the z-score to find how many standard deviations 17 is from the mean:
[tex]z = \frac{X - \mu}{\sigma}[/tex]
[tex]z = \frac{17 - 24}{7}[/tex]
[tex]z = \frac{-7}{7} = -1[/tex]
This result means 17 is 1 standard deviation below the mean.
- We calculate the z-score to find how many standard deviations 17 is from the mean:
Apply the 68-95-99.7 Rule:
- The 68-95-99.7 rule states:
- About 68% of data falls within 1 standard deviation of the mean.
- Therefore, 34% of the data lies between the mean and 1 standard deviation below (since it's symmetric).
- Since we're interested in the values above 17, we combine the 50% of data above the mean (24) and the 16% of data that is between the mean and one standard deviation below.
- The 68-95-99.7 rule states:
Calculate the Percentage Above 17:
- Percentage of data above the mean (24) is 50%.
- Percentage of data between 17 and 24 (which is 1 standard deviation below the mean) is 34%.
- Therefore, the percentage of values above 17:
[tex]50\% + 34\% = 84\%[/tex]
So, 84% of the values are above 17 in this normal distribution.
