https://file.aiquickdraw.com/mathbot/file/1728307843467di7ltc9p.jpg

Let's tackle these questions step by step using the Empirical Rule (also known as the 68-95-99.7 Rule).

### Question 9:
We are given the following information:
- Mean (μ) = 37 ounces
- Standard deviation (σ) = 9 ounces
- The distribution is normal (bell-shaped).

#### a) 68% of the widget weights lie between ____ and ____.
According to the Empirical Rule, 68% of the data lies within 1 standard deviation of the mean. So, the range is:
\[
\text{Mean} - 1\sigma \text{ to } \text{Mean} + 1\sigma
\]
\[
37 - 9 \text{ to } 37 + 9 \Rightarrow 28 \text{ to } 46 \text{ ounces}
\]
**Answer: 28 and 46 ounces**

#### b) What percentage of widget weights lie between 19 and 46 ounces?
Now, we calculate how many standard deviations the values 19 and 46 are from the mean:
\[
\frac{19 - 37}{9} = -2 \quad (\text{2 standard deviations below the mean})
\]
\[
\frac{46 - 37}{9} = 1 \quad (\text{1 standard deviation above the mean})
\]
According to the Empirical Rule:
- From 2 standard deviations below the mean to 1 standard deviation above the mean includes 81.5% of the data.
\[
P(19 \leq X \leq 46) = 81.5\%
\]
**Answer: 81.5%**

#### c) What percentage of the widget weights lie below 64?
Calculate how many standard deviations 64 is from the mean:
\[
\frac{64 - 37}{9} = 3 \quad (\text{3 standard deviations above the mean})
\]
According to the Empirical Rule:
- 99.7% of the data lies within 3 standard deviations of the mean. 
- Half of this (below the mean) is 49.85%, and everything above that is the remaining part:
\[
P(X \leq 64) = 99.85\%
\]
**Answer: 99.85%**

### Question 10:
We are given the following:
- Mean (μ) = 6.7 inches
- Standard deviation (σ) = 0.8 inches
- We want the range that covers 99.7% of the data, which is 3 standard deviations from the mean.

\[
\text{Mean} - 3\sigma \text{ to } \text{Mean} + 3\sigma
\]
\[
6.7 - 3(0.8) \text{ to } 6.7 + 3(0.8) \Rightarrow 4.3 \text{ to } 9.1 \text{ inches}
\]
**Answer: 4.3 and 9.1 inches**

---

Would you like more details on any of the steps, or do you have any further questions?

### Follow-up Questions:
1. How does the Empirical Rule help us approximate probabilities in a normal distribution?
2. What is the meaning of "standard deviation" in the context of this problem?
3. How would the percentages change if the distribution were skewed instead of normal?
4. Can we use the Empirical Rule for distributions that aren't bell-shaped?
5. What other rules or approximations can we use for understanding distributions?

### Tip:
Always sketch a quick normal distribution curve when using the Empirical Rule to visualize the percentages within 1, 2, and 3 standard

The Acme Company manufactures widgets. The distribution of widget weights is bell-shaped. The widget weights have a mean of 37 ounces and a standard deviation of 9 ounces. Use the Standard Deviation Rule, also known as the Empirical Rule. Suggestion: sketch the distribution in order to answer these questions: a) 68% of the widget weights lie between _____ and _____. b) What percentage of the widget weights lie between 19 and 46 ounces? c) What percentage of the widget weights lie below 64? Additionally, the breadth of male heads has a mean of 6.7 inches and a standard deviation of 0.8 inches. Use the 68-95-99.7 rule to find the range covering 99.7% of the head breadths.

Explore how the Empirical Rule applies to normally distributed data, such as widget weights with a mean of 37 ounces and a standard deviation of 9 ounces. Learn how to calculate percentages of data within certain ranges, including real-life applications like head breadth measurements for helmet design. Suitable for high school students, Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation