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Suppose that the mean heart rate in the U.S. is 70 beats per minute, with a standard deviation of 15 beats per minute. Use this information and the 68-95-99.7 rule to answer all parts below.

**Question 5**

**Part 1**: What percentage of heart rates are between 70 and 85 beats per minute? Write your answer as a percent without the percent symbol.

Answer :

Sure! Let's solve the problem step-by-step.

Given information:
- Mean heart rate = 70 beats per minute
- Standard deviation = 15 beats per minute

Question:
What percentage of heart rates are between 70 and 85 beats per minute?

Step-by-step solution:

1. Identify the Mean and Standard Deviation:
- Mean (μ) = 70 beats per minute
- Standard Deviation (σ) = 15 beats per minute

2. Determine the Boundaries:
- Lower bound (L) = 70 beats per minute
- Upper bound (U) = 85 beats per minute

3. Calculate the Z-scores for the Boundaries:
A Z-score measures how many standard deviations an element is from the mean.

[tex]\( Z_{lower} = \frac{L - \mu}{\sigma} = \frac{70 - 70}{15} = 0 \)[/tex]

[tex]\( Z_{upper} = \frac{U - \mu}{\sigma} = \frac{85 - 70}{15} = 1 \)[/tex]

4. Apply the 68-95-99.7 Rule:
The 68-95-99.7 rule (also known as the Empirical Rule) states:
- About 68% of the data falls within 1 standard deviation (σ) of the mean (μ).
- About 95% falls within 2σ.
- About 99.7% falls within 3σ.

Since 85 beats per minute is 1 standard deviation above the mean (70 + 15 = 85):
- The percentage of heart rates within 1 standard deviation from the mean (70 ± 15) is 68%.

5. Find the Percentage between 70 and 85:
- We are interested in the percentage of heart rates between 70 and 85 beats per minute.
- This range represents half of the interval between the mean and one standard deviation above the mean.

- Since 68% of the data lies within 1 standard deviation (from 55 to 85), half of this (from 70 to 85) would be:
[tex]\( \frac{68}{2} = 34 \% \)[/tex]

Final Answer:
34

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