https://file.aiquickdraw.com/mathbot/file/1728260168850qvtk7pms.jpg

Let's break down and solve the questions based on the given information.

### Given Data:
- Mean \( \mu = 9.12 \) ounces
- Standard deviation \( \sigma = 0.05 \) ounces
- Normal distribution is assumed, and we are using the empirical rule (68-95-99.7 rule).

#### (a) What percent of bags weigh less than 9.07 ounces?

To solve this:
- First, find how many standard deviations 9.07 ounces is from the mean using the z-score formula:
  \[
  z = \frac{x - \mu}{\sigma}
  \]
  where \( x = 9.07 \), \( \mu = 9.12 \), and \( \sigma = 0.05 \).
  
  Substituting the values:
  \[
  z = \frac{9.07 - 9.12}{0.05} = \frac{-0.05}{0.05} = -1
  \]

  A z-score of -1 corresponds to being 1 standard deviation below the mean. According to the empirical rule, approximately 68% of the data lies within 1 standard deviation of the mean, meaning 34% is below the mean and 34% is above the mean. Since the total area below the mean is 50%, and another 34% from 9.12 ounces to 9.07 ounces is above that point, we subtract the 34% from 50% to find the percentage below 9.07 ounces:
  \[
  \text{Percentage} = 50\% - 34\% = 16\%
  \]

  Therefore, **16% of the bags weigh less than 9.07 ounces**.

#### (b) What proportion of bags weigh between 8.97 and 9.17 ounces?

- Find the z-scores for 8.97 and 9.17 ounces.

For \( x = 8.97 \):
\[
z = \frac{8.97 - 9.12}{0.05} = \frac{-0.15}{0.05} = -3
\]

For \( x = 9.17 \):
\[
z = \frac{9.17 - 9.12}{0.05} = \frac{0.05}{0.05} = 1
\]

According to the empirical rule:
- Approximately 99.7% of the data falls within 3 standard deviations of the mean.
- Approximately 68% of the data falls within 1 standard deviation of the mean.

Since the z-scores are -3 and 1, the area between these values would be approximately 68% (for the range within ±1 standard deviation) plus a tiny fraction for the area between -3 and -1 (since nearly all the data is within 3 standard deviations).

Thus, the **proportion is approximately 84%** or **0.84** in decimal form.

Let me know if you'd like additional details or clarification on any part!

---

Here are 5 related questions to deepen your understanding:
1. How would you calculate the proportion for a z-score of -2 or +2?
2. What is the probability that a bag weighs more than 9.12 ounces?
3. How is the empirical rule useful for approximating probabilities in a normal distribution?
4. Can you find the exact percentage using a z-table instead of the empirical rule?
5. How does the empirical rule change for non-normal distributions?

**Tip:** The empirical rule is an approximation; for more precise answers, you can use a standard normal (z) table or statistical software.

The weights of 9-ounce bags of potato chips follow a normal distribution with a mean of 9.12 ounces and a standard deviation of 0.05 ounces. (a) What percent of bags weigh less than 9.07 ounces? (b) What proportion of bags weigh between 8.97 and 9.17 ounces?

Use the empirical rule to solve two questions about potato chip bag weights: (a) the percent of bags weighing less than 9.07 ounces, and (b) the proportion of bags weighing between 8.97 and 9.17 ounces, with a mean of 9.12 and standard deviation of 0.05.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation