The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 50 ounces and a standard deviation of 10 ounces.
Use the Standard Deviation Rule, also known as the Empirical Rule.
Suggestion: sketch the distribution in order to answer these questions.
a) 99.7% of the widget weights lie between
and
b) What percentage of the widget weights lie between 30 and 80 ounces?
%
c) What percentage of the widget weights lie below 60 ?

The **Empirical Rule**, also known as the **68-95-99.7 Rule**, applies to normally distributed data. It gives us approximate percentages of data that fall within 1, 2, and 3 standard deviations from the mean.

We know that the widget weights are normally distributed with:

- Mean (\(\mu\)) = 50 ounces
- Standard deviation (\(\sigma\)) = 10 ounces

### (a) 99.7% of the widget weights lie between:
- **99.7%** of the data lies within **3 standard deviations** from the mean.
  
  The range is calculated as:
  \[
  \text{Lower bound} = \mu - 3\sigma = 50 - 3(10) = 50 - 30 = 20 \text{ ounces}
  \]
  \[
  \text{Upper bound} = \mu + 3\sigma = 50 + 3(10) = 50 + 30 = 80 \text{ ounces}
  \]

So, **99.7% of the widget weights lie between 20 and 80 ounces**.

### (b) What percentage of the widget weights lie between 30 and 80 ounces?
- **30 ounces** is 2 standard deviations below the mean: 
  \[
  \text{Number of deviations from mean} = \frac{30 - 50}{10} = -2
  \]
- **80 ounces** is 3 standard deviations above the mean: 
  \[
  \text{Number of deviations from mean} = \frac{80 - 50}{10} = +3
  \]
  
According to the Empirical Rule:
- From \(-2\sigma\) to \(+3\sigma\) covers **from 30 to 80 ounces**.
- The percentage between \(-2\sigma\) and \(+3\sigma\) can be found by combining:
  - 95% of data is between \(-2\sigma\) and \(+2\sigma\) (i.e., 30 to 70 ounces)
  - From \(+2\sigma\) to \(+3\sigma\) covers an additional 2.35% of data.

Thus, the percentage of widget weights between 30 and 80 ounces is **95% + 2.35% = 97.35%**.

### (c) What percentage of the widget weights lie below 60 ounces?
- **60 ounces** is 1 standard deviation above the mean:
  \[
  \frac{60 - 50}{10} = +1
  \]
  
According to the Empirical Rule, **68%** of the data lies within 1 standard deviation of the mean, so:
- From \(50 - 10 = 40\) to \(50 + 10 = 60\), 68% of data lies between 40 and 60 ounces.
- Since the distribution is symmetric, 34% of the data lies between the mean and 60 ounces, and half of the data (50%) is below the mean.

Thus, the total percentage below 60 ounces is:
\[
50\% + 34\% = 84\%
\]

### Summary:
- **(a)** 99.7% of the widget weights lie between **20 and 80 ounces**.
- **(b)** 97.35% of the widget weights lie between **30 and 80 ounces**.
- **(c)** 84% of the widget weights lie **below 60 ounces**.

Would you like further details or explanations?

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Here are five related questions you can explore:
1. What percentage of widget weights lie between 40 and 60 ounces?
2. What percentage of widget weights lie above 70 ounces?
3. If the mean widget weight increased to 55 ounces, how would this affect the distribution?
4. What is the probability that a randomly chosen widget weighs more than 80 ounces?
5. How would the percentage of widget weights change if the standard deviation was reduced to 5 ounces?

**Tip**: The Empirical Rule is a quick approximation for normal distributions, but more precise calculations can be done using a Z-score table.

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 50 ounces and a standard deviation of 10 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule. a) 99.7% of the widget weights lie between and b) What percentage of the widget weights lie between 30 and 80 ounces? c) What percentage of the widget weights lie below 60?

Explore how to apply the Empirical Rule to a normally distributed data set with a mean of 50 ounces and a standard deviation of 10 ounces. This problem includes calculating the range for 99.7% of widget weights, finding the percentage of widgets between 30 and 80 ounces, and determining the percentage of widgets below 60 ounces. Suitable for students in Grades 9-12.

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