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To solve this problem, we need to calculate the probability that the average length \( M \) of a randomly selected bundle of 29 steel rods is less than 234.8 cm. Given that the lengths of the steel rods are normally distributed with a mean \( \mu = 235.5 \) cm and a standard deviation \( \sigma = 2.1 \) cm, we can proceed by finding the z-score for 234.8 cm.

### Step-by-Step Solution

1. **Determine the sampling distribution for the mean of a sample of 29 rods:**
   - Since the sample size \( n = 29 \), the standard error (SE) of the mean can be calculated as:
     \[
     \text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{2.1}{\sqrt{29}}
     \]

2. **Calculate the standard error:**
   \[
   \text{SE} = \frac{2.1}{\sqrt{29}} \approx 0.3895
   \]

3. **Calculate the z-score:**
   - The z-score for \( M = 234.8 \) cm is:
     \[
     z = \frac{M - \mu}{\text{SE}} = \frac{234.8 - 235.5}{0.3895} \approx -1.797
     \]

4. **Find the probability \( P(M < 234.8) \):**
   - Using the z-score table or a standard normal distribution calculator, we find the probability corresponding to \( z = -1.797 \).
   - This gives:
     \[
     P(M < 234.8) \approx 0.0362
     \]

### Final Answer
\[
P(M < 234.8) \approx 0.0362
\]

Would you like more details on any part of the process, or have further questions?

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### Related Questions:
1. How would the probability change if the sample size increased to 50 rods?
2. What is the probability that the average length is greater than 235.5 cm?
3. How do you calculate probabilities for a non-normal distribution?
4. What would the standard error be if the standard deviation was 3 cm instead?
5. How does changing the mean length affect the probability?

### Tip:
When calculating probabilities involving sample means, always consider the sample size because it affects the standard error, making the distribution of the sample mean narrower as the sample size increases.

A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 235.5 cm and a standard deviation of 2.1 cm. For shipment, 29 steel rods are bundled together. Find the probability that the average length of a randomly selected bundle of steel rods is less than 234.8 cm.

Learn how to calculate the probability that the average length of a bundle of 29 steel rods is less than 234.8 cm. Given a mean of 235.5 cm and standard deviation of 2.1 cm, this problem demonstrates the use of sampling distribution, z-score calculations, and normal distribution concepts.

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