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To solve for \( P_{24} \), which represents the length at the 24th percentile in a normal distribution, we need to use the following information given in the problem:

- **Mean** (\( \mu \)) = 91.6 cm
- **Standard Deviation** (\( \sigma \)) = 1.5 cm
- We are looking for the 24th percentile, so \( P_{24} \) is at the point where 24% of the distribution is below it.

### Step 1: Find the Z-score for the 24th Percentile
For a normal distribution, we can find the Z-score corresponding to the 24th percentile using a Z-table or standard statistical software. 

The Z-score for the 24th percentile is approximately:
\[
Z_{24} \approx -0.706
\]

### Step 2: Convert the Z-score to the Corresponding Length
The Z-score formula for a normal distribution is:
\[
Z = \frac{X - \mu}{\sigma}
\]
Rearranging this to solve for \( X \) (the value at the 24th percentile):
\[
X = Z \cdot \sigma + \mu
\]

Plugging in the values:
\[
X = (-0.706) \cdot 1.5 + 91.6
\]

Calculating this:
\[
X \approx -1.059 + 91.6 = 90.541
\]

### Answer
\[
P_{24} \approx 90.54 \text{ cm}
\]

Would you like a more detailed explanation or have any questions?

Here are 5 related questions:
1. How would you calculate the 76th percentile instead of the 24th?
2. What would change if the sample size affected the standard deviation?
3. How is the Z-score different for other percentiles?
4. Why do we use a Z-score table in normal distribution calculations?
5. How does this method change if we have a skewed distribution instead of a normal one?

**Tip:** Always check if the distribution type is normal before using Z-scores directly; this method applies specifically to normal distributions.

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 91.6 cm and a standard deviation of 1.5 cm. For shipment, 26 steel rods are bundled together. Find P24, which is the average length separating the smallest 24% bundles from the largest 76% bundles.

This problem demonstrates how to calculate the 24th percentile for the lengths of steel rods that are normally distributed with a mean of 91.6 cm and a standard deviation of 1.5 cm. Learn how to find P24 using Z-scores and percentile calculations, perfect for students in Grades 10-12.

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