Use the Empirical Rule. The mean speed of a sample of vehicles along a stretch of highway is 66 miles per​ hour, with a standard deviation of 5 miles per hour. Estimate the percent of vehicles whose speeds are between 56 miles per hour and 76 miles per hour.​ (Assume the data set has a​ bell-shaped distribution.)

The Empirical Rule (or the 68-95-99.7 rule) applies to bell-shaped (normal) distributions and helps estimate the percentage of data within certain standard deviations from the mean.

Given:
- Mean speed \( \mu = 66 \) miles per hour
- Standard deviation \( \sigma = 5 \) miles per hour
- We are asked to find the percentage of vehicles whose speeds are between 56 mph and 76 mph.

### Steps:
1. **Calculate how far 56 mph and 76 mph are from the mean**:
   - 56 mph is 10 miles per hour below the mean: 
     \[
     66 - 56 = 10 \text{ mph}
     \]
   - 76 mph is 10 miles per hour above the mean:
     \[
     76 - 66 = 10 \text{ mph}
     \]
2. **Convert these differences into standard deviations**:
   \[
   \frac{10}{5} = 2 \text{ standard deviations}
   \]
   So, 56 mph is 2 standard deviations below the mean, and 76 mph is 2 standard deviations above the mean.

3. **Apply the Empirical Rule**:
   - According to the rule, approximately **95%** of the data falls within 2 standard deviations from the mean in a normal distribution.
   
Thus, we can estimate that **95%** of vehicles have speeds between 56 mph and 76 mph.

Would you like more details or have any questions?

Here are 5 related questions:
1. What percentage of vehicles have speeds between 61 mph and 71 mph?
2. How do we calculate percentages for non-whole number standard deviations?
3. What does the Empirical Rule tell us about data points beyond 3 standard deviations?
4. How would the estimate change if the standard deviation were smaller?
5. Can the Empirical Rule be applied to non-normal distributions?

**Tip:** The Empirical Rule is very useful for quick approximations, but precise calculations can be made using Z-scores and normal distribution tables.

Use the Empirical Rule to estimate the percentage of vehicles traveling between 56 and 76 mph on a highway with a mean speed of 66 mph and a standard deviation of 5 mph. Understand key statistical concepts like normal distribution and standard deviation. Suitable for Grades 9-12.

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