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The problem presented uses the **Empirical Rule** (or 68-95-99.7 rule) for a normal distribution. The widget weights are normally distributed with a **mean of 43 ounces** and a **standard deviation of 10 ounces**.

### Empirical Rule Breakdown:
- 68% of the data lies within **1 standard deviation** (mean ± 1σ)
- 95% lies within **2 standard deviations** (mean ± 2σ)
- 99.7% lies within **3 standard deviations** (mean ± 3σ)

#### Part a) "95% of the widget weights lie between ___ and ___"

To find the range for 95% of the dataThe problem presented uses the **Empirical Rule** (or 68-95-99.7 rule) for a normal distribution. The widget weights are normally distributed with a **mean of 43 ounces** and a **standard deviation of 10 ounces**.

### Empirical Rule Breakdown:
- 68% of the data lies within **1 standard deviation** (mean ± 1σ)
- 95% lies within **2 standard deviations** (mean ± 2σ)
- 99.7% lies within **3 standard deviations** (mean ± 3σ)

#### Part a) "95% of the widget weights lie between ___ and ___"

To find the range for 95% of the 
- Use **2 standard deviations** from the mean:
  - Lower bound = Mean - 2σ = 43 - 2(10) = 43 - 20 = **23 ounces**
  - Upper bound = Mean + 2σ = 43 + 2(10) = 43 + 20 = **63 ounces**

Thus, 95% of the widget weights lie between **23 ounces and 63 ounces**.

#### Part b) "What percentage of the widget weights lie between 13 and 63 ounces?"

To answer this:
- Lower bound: 13 is **3 standard deviations below** the mean (43 - 3(10) = 13).
- Upper bound: 63 is **2 standard deviations above** the mean (calculated above).

From the Empirical Rule:
- From **-3σ to +2σ** contains:
  - 99.7% of the data lies between -3σ and +3σ.
  - Since we're considering only up to +2σ, we subtract the last 0.15% of the tail beyond +2σ.
  - So, the percentage of data between 13 and 63 ounces is **97.5%**.

### Final answers:
- a) 95% of the widget weights lie between **23 and 63 ounces**.
- b) **97.5%** of the widget weights lie between 13 and 63 ounces.

Would you like more details or have any questions about these calculations?

Here are 5 additional questions to deepen understanding:
1. How would the percentages change if the standard deviation increased to 15 ounces?
2. What does the Empirical Rule tell us about data points beyond 3 standard deviations from the mean?
3. Can the normal distribution have negative values in this context? Why or why not?
4. How would the range of weights differ if the percentage was 68% instead of 95%?
5. How do we calculate probabilities for non-standard deviations?

**Tip:** The Empirical Rule is a great approximation for normal distributions, but for more precision, especially for non-integer standard deviations, using the Z-score tables is recommended.

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 43 ounces and a standard deviation of 10 ounces. a) 95% of the widget weights lie between ___ and ___. b) What percentage of the widget weights lie between 13 and 63 ounces?

Solve the widget weights distribution problem using the Empirical Rule. Learn how 95% of the weights fall between 23 and 63 ounces and what percentage lies between 13 and 63 ounces using standard deviations.

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