The blood platelet counts of a group of women have a bell-shaped distribution with a mean of 246.6 and a standard deviation of 68.6 . (All units are 1000 cells $$ / \mu \mathrm{L} $$.) Using the eml percentage below. a. What is the approximate percentage of women with platelet counts within 2 standard deviations of the mean, or between 109.4 and 383.8 ? b. What is the approximate percentage of women with platelet counts between 40.8 and 452.4 ?

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To determine the approximate percentages of women with specific platelet counts, we'll apply the **Empirical Rule** (also known as the **68-95-99.7 rule**) for normally distributed (bell-shaped) data. This rule provides approximate percentages of data within 1, 2, and 3 standard deviations from the mean.

### Given:
- **Mean (μ):** 246.6 (in 1000 cells/µL)
- **Standard Deviation (σ):** 68.6 (in 1000 cells/µL)

### a. Percentage Within 2 Standard Deviations (109.4 to 383.8)

**Calculation:**
- **Lower Bound:** μ - 2σ = 246.6 - (2 × 68.6) = 246.6 - 137.2 = **109.4**
- **Upper Bound:** μ + 2σ = 246.6 + (2 × 68.6) = 246.6 + 137.2 = **383.8**

**Empirical Rule Interpretation:**
- Approximately **95%** of the data falls within **±2σ** of the mean.

**Answer:**
Approximately **95%** of women have platelet counts between **109.4** and **383.8**.

### b. Percentage Within 3 Standard Deviations (40.8 to 452.4)

**Calculation:**
- **Lower Bound:** μ - 3σ = 246.6 - (3 × 68.6) = 246.6 - 205.8 = **40.8**
- **Upper Bound:** μ + 3σ = 246.6 + (3 × 68.6) = 246.6 + 205.8 = **452.4**

**Empirical Rule Interpretation:**
- Approximately **99.7%** of the data falls within **±3σ** of the mean.

**Answer:**
Approximately **99.7%** of women have platelet counts between **40.8** and **452.4**.

### Summary
- **a.** ~95%
- **b.** ~99.7%
Approximately 95% of women have platelet counts between 109.4 and 383.8, and about 99.7% have counts between 40.8 and 452.4.

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