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The question involves calculating how many peaches are expected to have weights between 12.4 and 13.2 ounces, given a normally distributed sample with a mean of 12 ounces and a standard deviation of 0.8 ounces, with 1000 peaches in total. 

To solve this, we need to:
1. Convert the weight limits (12.4 ounces and 13.2 ounces) into z-scores using the formula:
   \[
   z = \frac{x - \mu}{\sigma}
   \]
   where \(x\) is the data point, \(\mu\) is the mean (12 ounces), and \(\sigma\) is the standard deviation (0.8 ounces).

2. Use the z-scores to find the corresponding probabilities from the standard normal distribution table provided in the image.

3. Subtract the lower probability from the upper probability to find the percentage of peaches in that range.

4. Multiply this percentage by 1000 to find the expected number of peaches in that range.

Let's calculate:

1. **For 12.4 ounces**:
   \[
   z = \frac{12.4 - 12}{0.8} = 0.5
   \]
   Looking at the z-table in the image for \(z = 0.50\), the area is approximately **0.6915**.

2. **For 13.2 ounces**:
   \[
   z = \frac{13.2 - 12}{0.8} = 1.5
   \]
   From the z-table, the area for \(z = 1.50\) is approximately **0.9332**.

3. The proportion of peaches between 12.4 and 13.2 ounces is:
   \[
   0.9332 - 0.6915 = 0.2417
   \]
   So, about **24.17%** of the peaches are expected to have weights in this range.

4. Finally, the number of peaches expected in this range is:
   \[
   0.2417 \times 1000 = 241.7 \approx 242
   \]

So, about **242 peaches** are expected to have weights between 12.4 and 13.2 ounces.

Would you like more details or have any further questions?

Here are 5 related questions:
1. How would the result change if the sample size was increased to 5000 peaches?
2. What is the z-score formula, and how does it apply to normal distributions?
3. How do you interpret the area under the standard normal curve?
4. Can you explain how changing the mean affects the distribution of the data?
5. How would the result differ if the standard deviation were larger or smaller?

**Tip**: Always round your final answer to match the precision required by the problem, especially when working with z-scores and probabilities.

The number of peaches expected to have weights between 12.4 and 13.2 ounces in a sample of 1000 peaches, where the weights are normally distributed with a mean of 12 ounces and a standard deviation of 0.8 ounces.

Learn how to calculate the number of peaches expected to weigh between 12.4 and 13.2 ounces from a sample of 1000 peaches. This problem covers normal distribution concepts, z-scores, and probability. Suitable for students in Grades 9-12.

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