Answer :
The question involves understanding the empirical rule, or the 68-95-99.7 rule, which applies to normal distributions. A distribution that is approximately bell-shaped, like this one, is often considered to be a normal distribution.
The empirical rule states the following about a normal distribution:
Approximately 68% of the data falls within one standard deviation (σ) of the mean.
Approximately 95% of the data falls within two standard deviations of the mean.
Approximately 99.7% of the data falls within three standard deviations of the mean.
Given:
- Mean (μ): 24.5
- Standard deviation (σ): 6.3
We need to find the proportion of values between 11.9 and 37.1.
Firstly, we find how many standard deviations away these values are from the mean.
Step 1: Calculate the standard deviations for the given values
For the lower value (11.9):
[tex]\frac{11.9 - 24.5}{6.3} = \frac{-12.6}{6.3} = -2[/tex]
For the upper value (37.1):
[tex]\frac{37.1 - 24.5}{6.3} = \frac{12.6}{6.3} = 2[/tex]
Step 2: Apply the empirical rule
According to the empirical rule, approximately 95% of the data should lie within two standard deviations of the mean.
Therefore, the proportion of values between 11.9 and 37.1 is approximately 95%.
So, based on the empirical rule, about 95% of the values in this dataset fall between 11.9 and 37.1.
