The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 61 ounces and a standard deviation of 9 ounces.

Use the Empirical Rule and a sketch of the normal distribution in order to answer these questions.

a) 95% of the widget weights lie between 
43
Correct and 
79
Correct

b) What percentage of the widget weights lie between 34 and 79 ounces? 
 %

c) What percentage of the widget weights lie below 70 ? 
84
Correct %

Given the information about the Acme Company's widget weights and the bell-shaped (normal) distribution, we can apply the **Empirical Rule** (also known as the 68-95-99.7 rule). This rule states that for a normal distribution:

- **68%** of the data lies within 1 standard deviation from the mean.
- **95%** of the data lies within 2 standard deviations from the mean.
- **99.7%** of the data lies within 3 standard deviations from the mean.

### a) 95% of the widget weights lie between:
To find the range that contains 95% of the widget weights, we calculate the values that are 2 standard deviations away from the mean:

- Mean \( \mu = 61 \) ounces
- Standard deviation \( \sigma = 9 \) ounces

**Lower bound**: 
\[
61 - 2 \times 9 = 61 - 18 = 43 \text{ ounces}
\]
**Upper bound**: 
\[
61 + 2 \times 9 = 61 + 18 = 79 \text{ ounces}
\]
So, **95%** of the widget weights lie between **43 ounces and 79 ounces**, which is already provided as correct.

### b) What percentage of the widget weights lie between 34 and 79 ounces?
To calculate this, we first express these values in terms of standard deviations from the mean:

- \( 34 \) ounces is:
\[
\frac{34 - 61}{9} = \frac{-27}{9} = -3 \sigma
\]
- \( 79 \) ounces is:
\[
\frac{79 - 61}{9} = \frac{18}{9} = 2 \sigma
\]

According to the Empirical Rule:
- From \( -3 \sigma \) to \( +3 \sigma \) contains **99.7%** of the data.
- From \( -2 \sigma \) to \( +2 \sigma \) contains **95%** of the data.

From \( -3 \sigma \) to \( +2 \sigma \) would contain \( 99.7\% - (100\% - 95\%) = 99.7\% - 5\% = 94.7\% \).

Thus, **94.7%** of the widget weights lie between **34 ounces and 79 ounces**.

### c) What percentage of the widget weights lie below 70 ounces?
First, we calculate how many standard deviations 70 is from the mean:

\[
\frac{70 - 61}{9} = \frac{9}{9} = 1 \sigma
\]

According to the Empirical Rule:
- **68%** of the data lies between \( -1 \sigma \) and \( +1 \sigma \), so from 52 to 70 ounces, **34%** lies between the mean (61 ounces) and \( +1 \sigma \) (70 ounces).

Since the normal distribution is symmetric, 50% of the data lies below the mean (61 ounces), and we add the 34% above the mean:

\[
50\% + 34\% = 84\%
\]

Thus, **84%** of the widget weights lie below **70 ounces**.

---

Would you like any further clarification or a more detailed explanation of any part of this?

### 5 Related Questions:
1. What percentage of the widget weights lie above 79 ounces?
2. What is the probability that a randomly chosen widget weighs less than 43 ounces?
3. What percentage of widget weights lie between 52 and 70 ounces?
4. How many standard deviations is a weight of 88 ounces from the mean?
5. What percentage of widget weights lie between 43 and 70 ounces?

### Tip:
When dealing with normal distributions, converting data points to "z-scores" (standard deviations from the mean) simplifies the process of calculating probabilities or percentages.

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 61 ounces and a standard deviation of 9 ounces. Use the Empirical Rule and a sketch of the normal distribution in order to answer these questions. a) 95% of the widget weights lie between 43 and 79 ounces. b) What percentage of the widget weights lie between 34 and 79 ounces? c) What percentage of the widget weights lie below 70?

Learn how to apply the Empirical Rule to a bell-shaped distribution of widget weights with a mean of 61 ounces and a standard deviation of 9 ounces. This problem covers calculating percentages of data points within standard deviations, using the normal distribution to find values between 34 and 79 ounces, and determining the percentage of weights below 70 ounces.

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