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------------------------------------------------ A data set with a distribution that is approximately bell-shaped has a mean of 24.5 and a standard deviation of 6.3. Use the empirical rule to determine the proportion of values between 11.9 and 37.1?

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.