The Empirical Rule (also known as the 68-95-99.7 Rule) is used to estimate the percentage of data that falls within certain standard deviations from the mean in a bell-shaped (normal) distribution. Specifically:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

Now, we have:

• Mean (μ) = 80
• Standard deviation (σ) = 5.659
• Scores between 63.023 and 91.318.

Step 1: Calculate the number of standard deviations from the mean

We need to determine how many standard deviations away 63.023 and 91.318 are from the mean.

Lower bound:

$z_{\text{lower}} = \frac{63.023 - 80}{5.659} = \frac{-16.977}{5.659} \approx -3.00$

Upper bound:

$z_{\text{upper}} = \frac{91.318 - 80}{5.659} = \frac{11.318}{5.659} \approx 2.00$

Step 2: Interpret the results

• The lower bound (63.023) is approximately 3 standard deviations below the mean.
• The upper bound (91.318) is approximately 2 standard deviations above the mean.

According to the Empirical Rule:

• About 95% of the data lies within 2 standard deviations from the mean (between $80 - 2\sigma$ and $80 + 2\sigma$).
• Nearly all the data (99.7%) lies within 3 standard deviations from the mean.

Step 3: Estimate the percentage

Since the range we are given is from 3 standard deviations below the mean to 2 standard deviations above the mean, this interval captures the majority of the data.

• From 3 standard deviations below the mean to the mean captures about 49.85% of the data.
• From the mean to 2 standard deviations above the mean captures 47.5% of the data.

Adding these gives approximately: $49.85\% + 47.5\% = 97.35\%$

Thus, about 97.35% of the individual scores are between 63.023 and 91.318.

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Would you like more details or have any questions? Here are some related questions:

1. How do we calculate exact percentages using the z-score table instead of the Empirical Rule?
2. Can the Empirical Rule be applied to all types of distributions?
3. What is the difference between the Empirical Rule and Chebyshev's Theorem?
4. How do we interpret the z-scores for more precise probability estimates?
5. How is the standard deviation used in other real-life scenarios like this?

Tip: The Empirical Rule is a quick estimation tool for normal distributions, but using z-scores and probability tables can provide more precise results.