A distribution of measurements is relatively mound-shaped with a mean of 80 and a standard deviation of 15. Use this information to find the proportion of measurements in the given interval.
between 65 and 95

We can solve this problem using the **Empirical Rule** (also known as the **68-95-99.7 rule**), which applies to mound-shaped (normal) distributions.

The Empirical Rule states:
- About **68%** of the data falls within 1 standard deviation (σ) of the mean (μ).
- About **95%** of the data falls within 2 standard deviations of the mean.
- About **99.7%** of the data falls within 3 standard deviations of the mean.

The given values are:
- Mean (μ) = 80
- Standard deviation (σ) = 15

The interval provided is between 65 and 95. To see how many standard deviations these values are from the mean:
- \( 65 = 80 - 15 \) (this is 1 standard deviation below the mean)
- \( 95 = 80 + 15 \) (this is 1 standard deviation above the mean)

Since both 65 and 95 are 1 standard deviation away from the mean, we can conclude that the proportion of measurements between 65 and 95 is approximately **68%**, based on the Empirical Rule.

### Conclusion:
The proportion of measurements between 65 and 95 is **68%**.

Would you like further details on the empirical rule or have any other questions?

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Here are some related questions to explore:
1. What is the proportion of measurements between 50 and 110?
2. How would the proportion change if the standard deviation was 10 instead of 15?
3. What is the probability of a measurement falling below 65?
4. What is the proportion of measurements greater than 95?
5. How does the empirical rule apply to skewed distributions?

**Tip**: The empirical rule works best for normally distributed data, but can still give approximate results for mound-shaped distributions.

A distribution of measurements is relatively mound-shaped with a mean of 80 and a standard deviation of 15. Use this information to find the proportion of measurements in the given interval: between 65 and 95.

Discover how to apply the Empirical Rule to find the proportion of measurements between 65 and 95 in a mound-shaped distribution with a mean of 80 and a standard deviation of 15. Learn key concepts in statistics such as the 68-95-99.7 rule. Ideal for Grades 9-12.

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