The body temperatures of a group of healthy adults have a​ bell-shaped distribution with a mean of

98.2698.26degrees°F

and a standard deviation of

0.610.61degrees°F.

Using the empirical​ rule, find each approximate percentage below.

**a.**

What is the approximate percentage of healthy adults with body temperatures within

11

standard

deviationdeviation

of the​ mean, or between

97.6597.65degrees°F

and

98.8798.87degrees°​F?

**b.**

What is the approximate percentage of healthy adults with body temperatures between

96.4396.43degrees°F

and

100.09100.09degrees°​F?

Question content area bottom

Part 1

**a.** Approximately

enter your response here​%

of healthy adults in this group have body temperatures within

11

standard

deviationdeviation

of the​ mean, or between

97.6597.65degrees°F

and

98.8798.87degrees°F.

​(Type an integer or a decimal. Do not​ round.)

The problem involves a bell-shaped distribution (which is a normal distribution) and uses the empirical rule (68-95-99.7 rule) to estimate percentages of data within certain ranges of standard deviations from the mean.

Given:
- Mean \( \mu = 98.26 \)°F
- Standard deviation \( \sigma = 0.61 \)°F

**a.** To find the approximate percentage of healthy adults with body temperatures within 1 standard deviation of the mean, between \( \mu - 1\sigma \) and \( \mu + 1\sigma \), we calculate:

\[
\mu - 1\sigma = 98.26 - 0.61 = 97.65 \text{°F}
\]

\[
\mu + 1\sigma = 98.26 + 0.61 = 98.87 \text{°F}
\]

Using the empirical rule:
- Approximately 68% of the data falls within 1 standard deviation of the mean in a normal distribution.

So, the approximate percentage is **68%**.

**b.** For temperatures between 96.43°F and 100.09°F, first, let's determine how many standard deviations these values are from the mean:

\[
96.43 \text{°F} = \mu - 3\sigma = 98.26 - 3(0.61) 
\]
\[
100.09 \text{°F} = \mu + 3\sigma = 98.26 + 3(0.61)
\]

The empirical rule tells us:
- About 99.7% of data falls within 3 standard deviations of the mean.

So, the approximate percentage is **99.7%**.

Do you want any additional details or have any questions? 

Here are some related questions:
1. What is the percentage of data within 2 standard deviations of the mean?
2. How does the empirical rule apply to non-normal distributions?
3. How do you calculate percentages for data more than 3 standard deviations away from the mean?
4. How can you use the z-score in normal distribution calculations?
5. What are some real-world examples where the empirical rule is used?

**Tip:** Always remember, the empirical rule is specifically for normal distributions. It may not apply to skewed or other non-normal distributions.

Explore how to estimate percentages within specified ranges of a bell-shaped distribution using the empirical rule. This problem involves calculating percentages of data within standard deviations from the mean, applicable to advanced high school or college-level mathematics.

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