aedy temperatures of a group of healthy adults have a bell-shaped distribution with a mean of $$ 98.29^{\circ} \mathrm{F} $$ and a standard deviation of $$ 0.58^{\circ} \mathrm{F} $$. Using the empirical rule, find a. What is the approximate percentage of healthy adults with body temperatures within 1 standard deviation of the mean, or between $$ 97.71^{\circ} \mathrm{F} $$ and $$ 98.87^{\circ} \mathrm{F} $$ ? b. What is the approximate percentage of healthy adults with body temperatures between $$ 97.13^{\circ} \mathrm{F} $$ and $$ 99.45^{\circ} \mathrm{F} $$ ?

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To determine the approximate percentages of healthy adults with body temperatures within specific ranges, we'll use the **Empirical Rule** (also known as the 68-95-99.7 rule) for normally distributed data:

1. **Within 1 Standard Deviation (σ) of the Mean (μ):** Approximately **68%** of the data falls within this range.
2. **Within 2 Standard Deviations (2σ) of the Mean (μ):** Approximately **95%** of the data falls within this range.
3. **Within 3 Standard Deviations (3σ) of the Mean (μ):** Approximately **99.7%** of the data falls within this range.

Given:
- **Mean (μ):** $$ 98.29^{\circ} \mathrm{F} $$
- **Standard Deviation (σ):** $$ 0.58^{\circ} \mathrm{F} $$

### a. **Within 1 Standard Deviation ($$ \mu \pm \sigma $$):**
- **Range:** $$ 98.29^{\circ} \mathrm{F} \pm 0.58^{\circ} \mathrm{F} $$ → **$$ 97.71^{\circ} \mathrm{F} $$ to $$ 98.87^{\circ} \mathrm{F} $$**
- **Approximate Percentage:** **68%**

### b. **Within 2 Standard Deviations ($$ \mu \pm 2\sigma $$):**
- **Range:** $$ 98.29^{\circ} \mathrm{F} \pm 2 	imes 0.58^{\circ} \mathrm{F} $$ → **$$ 97.13^{\circ} \mathrm{F} $$ to $$ 99.45^{\circ} \mathrm{F} $$**
- **Approximate Percentage:** **95%**

**Summary:**
- **a. Approximately 68%** of healthy adults have body temperatures between $$ 97.71^{\circ} \mathrm{F} $$ and $$ 98.87^{\circ} \mathrm{F} $$.
- **b. Approximately 95%** of healthy adults have body temperatures between $$ 97.13^{\circ} \mathrm{F} $$ and $$ 99.45^{\circ} \mathrm{F} $$.

**Answer:**  
a. Approximately 68 %  
b. Approximately 95 %
a. Approximately 68% of healthy adults have body temperatures between $$ 97.71^{\circ} \mathrm{F} $$ and $$ 98.87^{\circ} \mathrm{F} $$. b. Approximately 95% of healthy adults have body temperatures between $$ 97.13^{\circ} \mathrm{F} $$ and $$ 99.45^{\circ} \mathrm{F} $$.

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