mework The body temperatures of a group of healthy adults have a bell-shaped distribution with a mean of $$ 98.09^{\circ} \mathrm{F} $$ and a standard deviation of $$ 0.68^{\circ} \mathrm{F} $$. Using the empirical rule, find each approximate percentage below. a. What is the approximate percentage of healthy adults with body temperatures within 2 standard deviations of the mean, or between $$ 96.73^{\circ} \mathrm{F} $$ and $$ 99.45^{\circ} \mathrm{F} $$ ? b. What is the approximate percentage of healthy adults with body temperatures between $$ 97.41^{\circ} \mathrm{F} $$ and $$ 98.77^{\circ} \mathrm{F} $$ ?

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UpStudy AI · Accepted Answer

To solve these problems, we'll apply the **Empirical Rule** (also known as the **68-95-99.7 rule**), which applies to bell-shaped (normal) distributions.

The Empirical Rule states:
- **Approximately 68%** of the data lies within **1 standard deviation** of the mean.
- **Approximately 95%** lies within **2 standard deviations**.
- **Approximately 99.7%** lies within **3 standard deviations**.

Given:
- **Mean ($$ \mu $$)** = $$ 98.09^{\circ} \mathrm{F} $$
- **Standard Deviation ($$ \sigma $$)** = $$ 0.68^{\circ} \mathrm{F} $$

### a. Percentage within 2 Standard Deviations ($$ 96.73^{\circ} \mathrm{F} $$ to $$ 99.45^{\circ} \mathrm{F} $$)

**Calculation:**
- **Interval:** $$ \mu \pm 2\sigma = 98.09 \pm 2 \times 0.68 = [96.73, 99.45] $$
- **Percentage:** According to the Empirical Rule, **approximately 95%** of the data falls within this range.

**Answer:**  
**a.** Approximately **95%** of healthy adults have body temperatures between $$ 96.73^{\circ} \mathrm{F} $$ and $$ 99.45^{\circ} \mathrm{F} $$.

### b. Percentage between $$ 97.41^{\circ} \mathrm{F} $$ and $$ 98.77^{\circ} \mathrm{F} $$

**Calculation:**
1. **Determine the number of standard deviations each boundary is from the mean (z-scores):**
   - Lower Bound ($$ 97.41^{\circ} \mathrm{F} $$):
     $$
     z = \frac{97.41 - 98.09}{0.68} = -1
     $$
   - Upper Bound ($$ 98.77^{\circ} \mathrm{F} $$):
     $$
     z = \frac{98.77 - 98.09}{0.68} = 1
     $$
2. **Interpretation:** The range from $$ 97.41^{\circ} \mathrm{F} $$ to $$ 98.77^{\circ} \mathrm{F} $$ corresponds to **$$ \mu \pm 1\sigma $$**.
3. **Percentage:** According to the Empirical Rule, **approximately 68%** of the data lies within 1 standard deviation of the mean.

**Answer:**  
**b.** Approximately **68%** of healthy adults have body temperatures between $$ 97.41^{\circ} \mathrm{F} $$ and $$ 98.77^{\circ} \mathrm{F} $$.
**a.** Approximately **95%** of healthy adults have body temperatures between $$ 96.73^{\circ} \mathrm{F} $$ and $$ 99.45^{\circ} \mathrm{F} $$. **b.** Approximately **68%** of healthy adults have body temperatures between $$ 97.41^{\circ} \mathrm{F} $$ and $$ 98.77^{\circ} \mathrm{F} $$.

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