https://file.aiquickdraw.com/mathbot/file/1727441763619af4rr5g4.jpg

The problem in the image involves a normal distribution with the following given values:

- Mean (\(\mu\)) = 9.12 ounces
- Standard deviation (\(\sigma\)) = 0.05 ounces

We need to use the empirical rule (68-95-99.7 rule) to answer two questions.

### (a) What percent of bags weigh less than 9.07 ounces?
To solve this:

1. First, find the z-score for 9.07 ounces:
   \[
   z = \frac{x - \mu}{\sigma} = \frac{9.07 - 9.12}{0.05}
   \]

2. Then, we will use the z-score to determine the percentage.

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

1. Calculate the z-scores for 8.97 ounces and 9.17 ounces:
   \[
   z_1 = \frac{8.97 - 9.12}{0.05}
   \]
   \[
   z_2 = \frac{9.17 - 9.12}{0.05}
   \]

2. Use the z-scores to find the proportion of bags within this range.

Let me calculate these values.Here are the calculated z-scores:

- For part (a), the z-score for 9.07 ounces is approximately \(-1.00\).
- For part (b), the z-scores are:
  - For 8.97 ounces: \(-3.00\)
  - For 9.17 ounces: \(1.00\)

Now, I'll use these z-scores to find the respective percentages and proportions using the standard normal distribution. Let's proceed.Here are the results:

- **(a)** Approximately **15.87%** of the bags weigh less than 9.07 ounces.
- **(b)** The proportion of bags that weigh between 8.97 and 9.17 ounces is approximately **0.840**.

Would you like any additional details or clarifications?

Here are 5 related questions for further practice:
1. What percentage of bags weigh more than 9.17 ounces?
2. What is the probability of a bag weighing between 9.07 and 9.22 ounces?
3. What percentage of bags weigh less than 9.00 ounces?
4. What is the z-score for a bag weighing exactly 9.12 ounces?
5. If the standard deviation changed to 0.10 ounces, how would that affect the proportion between 8.97 and 9.17 ounces?

**Tip:** The z-score tells you how many standard deviations a value is from the mean. This is essential for converting between raw scores and probabilities in a normal distribution.

The weights of 9-ounce bags of a particular brand of potato chips can be modeled by a normal distribution with mean μ = 9.12 ounces and standard deviation σ = 0.05 ounce. (a) What percent of bags weigh less than 9.07 ounces? (b) What proportion of bags weigh between 8.97 and 9.17 ounces?

Solve a normal distribution problem involving 9-ounce bags of potato chips. Learn how to find the percentage of bags weighing less than 9.07 ounces and the proportion of bags weighing between 8.97 and 9.17 ounces using z-scores and the empirical rule.

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