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To solve this problem, we will use the **Central Limit Theorem** (CLT), which tells us that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the original distribution.

### Given:
- The population mean (\(\mu\)) is 209.6 cm.
- The population standard deviation (\(\sigma\)) is 0.5 cm.
- The sample size (\(n\)) is 19.

We need to find the probability that the average length of a randomly selected bundle of 19 steel rods is less than 209.6 cm.

The probability you're being asked to find is \(P(M < 209.6)\), where \(M\) represents the sample mean. To solve this, follow these steps:

1. **Calculate the standard deviation of the sample mean (standard error):**
   \[
   \sigma_M = \frac{\sigma}{\sqrt{n}} = \frac{0.5}{\sqrt{19}} \approx 0.1147
   \]

2. **Find the z-score** for \(M = 209.6\):
   The z-score is given by:
   \[
   z = \frac{M - \mu}{\sigma_M} = \frac{209.6 - 209.6}{0.1147} = 0
   \]

3. **Find the probability corresponding to the z-score**:
   For a z-score of 0, the probability is exactly 0.5.

Therefore, the probability that the average length of the steel rods is less than 209.6 cm is **0.5000**.

Would you like further details or have any questions?

Here are 5 related questions you can explore:
1. What is the central limit theorem and when is it used?
2. How would the probability change if the sample size increased?
3. What if the standard deviation of the rods was different, how would that affect the result?
4. How is the z-score affected if the mean of the population was different?
5. Can you find the probability for different sample mean values, such as \( P(M < 209.5) \)?

**Tip:** When dealing with the Central Limit Theorem, increasing the sample size reduces the standard error, which makes the sample mean more likely to be close to the population mean.

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 209.6 cm and a standard deviation of 0.5 cm. For shipment, 19 steel rods are bundled together. Find the probability that the average length of a randomly selected bundle of steel rods is less than 209.6 cm.

Learn how to calculate the probability that the average length of a randomly selected bundle of steel rods is less than 209.6 cm using the Central Limit Theorem. This problem involves key statistical concepts such as normal distribution, standard error, and z-scores.

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