The Empirical Rule, also known as the 68-95-99.7 rule, is used to approximate the percentage of data that falls within 1, 2, or 3 standard deviations from the mean in a normal distribution.

Given:

• Mean (μ) = 50
• Standard deviation (σ) = 15

We need to find the approximate percentage of data between 20 and 80.

Step-by-Step Solution:

1. Convert the values to z-scores (in terms of standard deviations):

   • $20$ is: $z_1 = \frac{20 - 50}{15} = \frac{-30}{15} = -2$
   • $80$ is: $z_2 = \frac{80 - 50}{15} = \frac{30}{15} = 2$

2. Apply the Empirical Rule:

   • According to the Empirical Rule:
     • About 68% of the data lies within 1 standard deviation of the mean.
     • About 95% of the data lies within 2 standard deviations of the mean.
     • About 99.7% of the data lies within 3 standard deviations of the mean.

3. Interpret the result:

   • Since both 20 and 80 are 2 standard deviations from the mean, about 95% of the data lies between these values.

Final Answer:

Approximately 95% of the data falls between 20 and 80.

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Would you like more details on this process or have any questions?

Here are five related questions to explore:

1. What percentage of data falls within 1 standard deviation from the mean?
2. How do you calculate a z-score for a given value?
3. What does it mean for data to be "normally distributed"?
4. What percentage of data is outside of 3 standard deviations from the mean?
5. How does the Empirical Rule compare to Chebyshev’s Theorem?

Tip: The Empirical Rule applies specifically to normal (bell-shaped) distributions. For non-normal distributions, other methods like Chebyshev’s Theorem are more appropriate.