The empirical rule, sometimes referred to as the 68-95-99.7 rule, can help us understand the percentages of data that lie within certain intervals for a normal distribution. According to this rule, approximately:
- 68% of the data falls within one standard deviation of the mean (mean ± 1 standard deviation),
- 95% falls within two standard deviations of the mean (mean ± 2 standard deviations),
- and 99.7% falls within three standard deviations of the mean (mean ± 3 standard deviations).
In this case, we are given a mean (μ) of 24 and a standard deviation (σ) of 5. We want to find the percentage of values that lie above 14.
Let's locate the value 14 in relation to the mean and the number of standard deviations it is away from the mean.
Distance from the mean:
[tex]\( 14 - 24 = -10 \)[/tex]
Numbers of standard deviations away from the mean at 14:
[tex]\( \frac{-10}{5} = -2 \)[/tex]
This result shows that 14 is 2 standard deviations less than the mean.
According to the empirical rule:
- Approximately 2.5% of the data falls more than 2 standard deviations below the mean (below [tex]\( \mu - 2\sigma \)[/tex]).
- If 2.5% of the data is below 14, then the remaining percentage will be above 14.
- To find this percentage, we subtract 2.5% from 100%.
Percentage of values above 14:
[tex]\( 100\% - 2.5\% = 97.5\% \)[/tex]
Therefore, approximately 97.5% of the values lie above 14 in this normal distribution.
